ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR TECHNICAL REPORT NO. 3 NEW OPERATIONAL MATHEMATICS THE OPERATIONAL CALCULUS OF LAGUERRE TRANSFORMS J. C. McCully R. V. Churchill Supervisor Project No. 2137 DETROIT ORDNANCE DISTRICT, DEPARTMENT OF THE ARMY CONTRACT NO. DA-20-018-ORD-12916 September, 1954

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN ABSTRACT Let e-x Ln(x) serve as the kernel function for a linear integral transformation, where Ln(x) is the Laguerre polynomial of nth degree. Operational properties, including a convolution property, are derived here. Transforms of particular functions as well as a few examples and applications are given.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN TABLE OF CONTENTS Page ABSTRACT ii I. INTRODUCTION 1. Integral Transforms 1 2. Properties of Laguerre Polynomials 2 3. Laguerre Kernels 4 II. LAGUERRE TRANSFORMS 1. The Transform 8 2. The Basic Operational Property 10 III. OPERATIONAL PROPERTIES 1. The Iterated Operator 13 2. Differentiation and Indefinite Integration 13 3. The Inverse Operator 15 4. Miscellaneous Remarks 16 IV. THE LAGUERRE CONVO)LDUTION 1. Introdu ction 18 2. The Addition Property 19 3. The Convolution Property 21 4. Remarks 25 V. TRANSFORMS OF PARTICULAR FUNCTIONS 1. Simple Transforms 33 2. Generating Functions and Laguerre Transforms 34 3- Prod.ucts of Transforms 35 4. Table of Laguerre Transforms 36 5. Table of Operational Properties 37 VI. EXAMPLES AND APPLICATIONS 1. Introduction 38 2, The Transform and Laguerre's Equation 38 3. Partial Differential Equations and the Laguerre Transform 42 - iii

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN TABLE OF CONTENTS (concl.) Page VII. SONINE TRANSFORMS 1. Introduction 46 2. Sonine Transforms 46 3. Properties of Sonine Polynomials 46 4. Operational Properties 47 BIBLIOGRAPHY 49 iv

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN TECHNICAL REPORT NO. 3 NEW OPERATIONAL MATHEMATICS THE OPERATIONAL CALCULUS OF LAGUERRE TRANSFORMS CHAPTER I INTRODUCTION 1. Integral Transforms When the function K(a,x) is a known function of the two variables a and x and the integral C (1) I(a) = F(x) K(a,x) dx is convergent, then the equation (1) defines a function of the variable a. This function is called the integral transform of the function F(x) by the kernel K(ax). One of the better known examples of such a kernel is (2) K(a,x) = e-ax which leads to the Laplace transform. Examples of other transforms can be found in Sneddon[14] and Tranter [17]. It follows immediately from the definition (1) that, if F(x) and G(x) are two functions which possess integral transforms by the kernel K(a,x) then the integral transform of their sum is (3) o [F(x) + G(x)] K(a,x) dx F(x) K(a,x) dx + G(x) K(a,x) dx. Jb Jb b

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN If d is a scalar, (4) + d F(x) K(a,x) dx = d F(x) K(a,x) dx. Equations (3) and (4) express the fact that the integral transform is a linear operator. We will direct our attention in what follows to a particular choice of the constants b, c, and the kernel function K(a,x) in the definition (1). We will choose b = O0 c = 00f and the kernel will involve a Laguerre polynomial. 2. Properties of Laguerre Polynomials We list here from the literature various properties of the Laguerre polynomials which will be of use to us later. Following Szego [16] we define the Laguerre polynomials Ln(x) by the following conditions of orthogonality: 00 0 if m f n JO 1 if m n We note here that Courant and Hilbert [5] denote by Ln(x) a function which is the same as n! Ln(x) in our notation. Laguerre uses the notation Fn(x) = n: Ln(-x). We have the differential equations xy" + (l-x) y' + ny = O0 Y = Ln(x), (6) xzt" + (l+x) z' + (n+l) z = 0, z = e-X Ln(x), and.x XU" + u. + (n+L X) u = O, u = e2 Ln(x). 2 4 The Rodriquesr formula for Laguerre polynomials is ex d n ~~(7) ~Ln(x) - n d (xn e-x).

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Other properties which will be of interest are as follows: (8) xLA(x) = nLn(x) - nLnl(x), (9) L(x) = L_ (x) -Ln (x) and (10) L(x) = - Lk(X) k=O It follows immediately from the relation n k (11) Ln(x) = nX ( x) (n-k)' (k, )2 k=O that the first few polynomials are Lo(x) = 1, L3(x) = 1-3x + -x (12) Ll(x) = l-x X2 X4 L2(x) = 1-2x +- 9 L4(x) = 1-4x + 3x2-_ - X3 + X 2 3 24 The Laguerre polynomials possess the following generating functions: 00, tn (13) et J~(2 4P = Ln1(x) n-' n=O where Jo(2 4'x) is the zero order Bessel function, (14) -exp Ln tn < 1. 1-t x1=O Of particular interest in connection with the convolution property of the Laguerre transforms will be the addition property (15) tLn(x) Ln(y) = 0 e-xy cos 0 cos ( xy sin G) Ln(x+y-2 e/y cos e,dQ. n~x) Ln~y0

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Relation (13) can be found in Rainville [131, the addition property (15) follows from a property on Sonine polynomials in Bateman [1], and the remaining properties can be found in Szego6 [16] or Erdelyi [6]. 3. Laguerre Kernels A kernel function K(a,x) will be called a Laguerre kernel if it involves a Laguerre polynomial. We will consider the three possible choices e Ln(x), e a Ln(x), and Ln(x) as Laguerre kernel functions. In this section we will attempt to show how the need for knowledge of properties of an integral transformation based on such a kernel function as one of the above might arise. Suppose one desired to obtain an advantageous resolution of the differential form (16) L[F(x)] = xF't(x) + (l-x) F'(x) into a simpler form-, The variable x will be allowed to range over the semi.-infinite interval from zero to infinity. We will now follow a procedure outlined by R. V. Churchill in 1950 in some unpublished notes. We will assume that the function F(x) has a continuous derivative of the second order with respect to x, x > O, and F(x) is Y(erX), r < 1, as x tends to infinity. We shall determine a kernel K(a,x) such that the linear integral transformation 00 (17) T[F(x)] = F(x) K(a,x) dx resolves the differential form L[F(x)] in terms of the transform T[F(x)]. Let us also assume that K(a,x) has a continuous derivative of the second order with respect to x, on the range x > 0. We assume the following form of the resolution: (18) T{L[F(x)]I = k(a) T[F(x)].

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN We will consider the self adjoint form of the differential form L[F(x)], To do this we write (19) r(x) = exp [ l-xdx], p(x) e x X x then L[F] = eX[xe-X F" + (l-x) e X F'] = eX(xe x F')'. We now write the kernel in terms of a new function M, K(a,x) = e'x M(a,x). The equation (17) now becomes 00 (20) T[F(x)] e x F(x) M(a,x) dx. We see that for F(x) of the order C(erX), r < 1 that the integral (20) will exist as long as M(a,x) does not become infinite of an order higher than a positive power of x. By successive integration by parts we can write T L[F] = f (xe-x F')' M(a,x) dx CJ ~~~~~~~~~~~~~~O~00 00 = f,0 (xe M' )'F dx + M(a,x) xe-x F'(x) 00 - M'xe-X F 0 In view of form (18) for Tt L[F]3 it follows inat 00 00 00 (21)f0 [(xe-X M')' - (a) e-x M] F dx = M'xe-x F - Mex x F If we assume that IMIFI < MI esx, M1 constant, s <1, as x + IMF < M2etX M2 constant, t < l,as x + 0, then the right hand side of equation (20) willbe equal to zero. Since the functions M(a,x), xeX, and A are independent of F it follows that ~ Jl:~~~~~~~~

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN (22) [xe-x Mt(ax)]' - X(a) e-x M(a,x) = 0 (0 < x < The equation (22) along with the conditions that M(ax) is bounded at the origin and does not become infinite of an order higher than a positive power of x make up a Sturm — Lio.uville system. The values of' for which this system has solutions that are not identically zero are the characteristic numbers [see the first one of equations (6)] -.x = An (n = 0, 1, 2,.2o) of this system. Courant and Hilbert [5] show that the characteristic numbers here are the negative integers X = -n,. The characteristic functions corresponding to these values of X are the so-called Laguerre polynomials, This family of characteristic functions is our kernel. The integral transformation (20) becomes (23) T[F(x)] = e-x Ln(xIF(x) dxF(n) (n=O, 1, 2,-, -); we shall call it the Laguerre transformation and f(n) represents the Laguerre transform of F(x). In view of equation (18) this transformation resolves the:form (16) as follows: (24) T fL[F(x)] = -n f(n), (n=O, 1, 2, *..). If we would have considered above the differential form [see eqs (6)] (25) L[F] = (xF) - F instead of the form (16) we would have arrived at the integral'transformation 00 x (26) T[F] = + e Ln(x) F(x) dx = f(n) (.n=0O 1, 2, *-.). This transformation is seen to resolve the form [25] as follows: (27) TtL[F] = - (n+ 1) T [F1, (n=O', 2,..). Application of the above process to the form[see equations (6)] (28) L*[F] = xF" + (l+x) F' + F leads to the integral transformation (29) T*[F] - 0 Ln(x) F(x) dx. 0

ENGINEERING RESEARCH INSTITUTE'I UNIVERSITY OF MICHIGAN The transformation (29) resolves the form (28) as follows: (30) T* {FE = -nf(n), (n=O, 1, 2,...). We notice in the three integral transformations (23), (26), and (29). that the function F(x) will have to satisfy different order properties in order for the transformation integral to exist. We will now abandon this approach and proceed by centering our attention on the kernel function e'x Ln(x) and we will derive various properties for a linear integral transformation built on this kernel function.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN CHAPTER II LAGUERRE TRANSFORMS 1. The Transform The sequence of numbers f(n) defined by the equation 00 (31) f(n) = F(x) e-X Ln(x) dx (n = 0, 1, 2,-*.) l where Ln(x) denotes the Laguerre polynomial of degree n, is the Laguerre transform of the transform F(x). The integral transformation here will be denoted by TLF(x)j. The Laguerre transform of a function F(x) exists if F(x) is sect tionally continuous in every finite interval in the range x > 0 and if the function is dY(eaX), a < 1 as x tends to infinity. Under the conditions stated, the integrand of the Laguerre integral is integrable over the finite interv:al 0 < x 1- xo for every positive number xo, and since Ln(x) does not become infinite of an order higher than a finite power of x Ie-x F(x) Ln(x) I < M e-bx, b > O, where M is some constant. The integral of the function on the right exists. Hence the Laguerre integral converges absolutely when a < 1.The inverse of this transformation is represented here by the expansion of F(x) in a series of the Laguerre polynomials. The inversion process here can be thought of as an expansion in an infinite series in terms of the eigenfunctions Ln(x). This differs from the case of continluous spectra where a Fourier integral theorem would replace the eigenfunction expansion. The inverse of the Laguerre transformation is then 00 (32) F(x) = X f(n) Ln(x) = T'- if(n)} (O < x < o).

- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Uspensky [181 gives the following conditions which will guarantee the convergence of the Laguerre series: let (1) f e-X [F(x)]2 dx exist for a certain constant a, (2) f x- IF(x)I dx exist for a certain value of b, (3) F(x) be of bounded variation in a certain interval x-d, x+d, and absolutely integrable in any finite interval; then -[F(x+O) - F(x-0)] = an Ln(x) 2 n=O where an = e'- Ln(x) F(x) dx. It is the necessity of taking into account the infinite values of the variable that constitutes the essential difficulty of the problem of the development of arbitrary functions in series of Laguerre polynomials. The first two conditions above take care of the difficulties brought into the problem in such a way. The summability of the series has been discussed by E. Hille [7] and G. Szego [16]. The Parseval theorem for the series has been investigated by S. Wigert [19]. Wigert [19] states the following theorem: "If the function F(x) is continuous for x = O0 and the integral 0 e-ax IF(x)i dx converges for a > 1 one has for x 0 O 2 0O lim f(n) Ln(x)rn = F(x)." ral n=O Wigert shows that the hypotheses given on F(x) imply limfljf(n)| < 1 which condition must be satisfied if -00 Lf(n) Ln(X) n=O is to be convergent. Wigert demonstrates that the integral condition is necessary by considering the function F(x) = ebx 1 > b > i. He shows the 9 ~'

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Laguerre series does not converge. This illustrates a situation which occurs in other linear integral transformations. The function F(x) ebx 1 > b > 1, has a transform, namely: T ~,bxf (_l-b ( -b) (1 > b> 1 -b (-b 2 but its Laguerre series does not converge. In other words conditions on a function insuring the inverse process are more severe than conditions necessary for the existence of the transform. It follows from the inequality e- |ILn(x) I < 1, Szeg8o [16], that ifZ If(n)l converges, then n=O f(n) Ln(X) n=O will converge and will represent the inverse transformation. That this condition is sufficient and not necessary c&n be seen from the expansion -- e2 = (-1 Ln(X). 2 n=O The above expansion is given in Wigert [19]. _2. The Basic Operational Property Let L[F] denote the differential form (33) ex [xe-x F']'. When the integral Tf L[F]t is integrated successively by parts and -nLn(x) is substituted for L[Ln(x)] in accordance with Laguerre's differential equation, the following result is easily obtained. Theorem 1: Let F(x) denote a function that satisfies these conditions: Ft'(x) is continuous and F"(x) is sectionally continuous over each finite interval contained in the range x ~ O; F(x) and Ft (x) are (eaX), a < 1,.___ __ __ __ __ __ __ __ __ _,10

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN as x tends to infinity. Then T {L[F(x)]t exists and (34) T fL[F(x)]t = -nf(n) (n=O, 1, 2,' ). Formula (34) represents the first basic operational property of the Laguerre transformation T under which the differential operation L[F] defined by equation (33) is replaced by the algebraic operation -nf(n). We note here that in deriving the first basic operational property we have in a sense reversed the:procedure used in section 3 when we were establishing the form of the kernel which would annihilate form (33). Relations (6) exhibited three forms of Laguerre's differential equation. We have seen in Theorem 1 the result of applying the Laguerre transform to the first of the three forms. In section 3 we obtained the kernels of integral transformations which would annihilate parts of the remaining forms in equation (6). -We will now investigate the result of applying the Laguerre transform to the second and third equations in expression (6). Let R[F] denote the differential form (35) e-x [xeX F''. Then by integration by parts we can write 00 00 TtR[F]j = f e-2X Ln(x) [,xeX F']' dx = - xF' e-X L+l1(x) dx 0 +J e- x xLn(x) F' dx; here we have used property (9) of Laguerre polynomials to write the first integral on the right. Integrating by parts again gives T fR[F] = f [xe-x LI+l(x)]' F(x) dx + e- X xLn(x) F'(x) dx. We can replace the expression [xe X Ln+l(x)]' by -(n+l) Ln+l(x) in the first integral on the right. In order to complete the derivation we must now find T {xF'?. By integration by parts we can write T {xF' = e-X xLn(x) F'(x) dx = - e-X[xLI(x) + (l-x) Ln(x)] F(x) dx......-_ _ _ _ _ _ _ 11.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Use of properties (8) and (9) leadsto the following: xL -(x) -xLn(x) = (n+l) Ln+l(x) - (n+l) Ln(x). Hence T ExF'} = - [ e-X [( Ln+l(x) - nLn(x)] F(x) dx = - (n+l) f(n+l) + nf(n). We can now write T JR[F]t as follows: T [R[F]t = -2(n+l) f(n+l) + nf(n). We summarize the above in the following theorems. Theorem 2: Let F(x) denote a function that satisfies these conditions: F' (x) is sectionally continuou.s over each finite interval in the range x = O F(x) is &(eax), a < 1, as x tends to infinity. Then T ~xF'l exists and (36) T fxF'} = -(n+l) f(n+l) + nf(n) (n=O, 1, 2,...). Theorem 3: Let F(x) denote a function that satisfies the conditions of Theorem 1. Then T el[F]~ exists and (37) T {R[F]} = -2(n+l) f(n+l) + f(n) (n=O, 1, 2,...). We note in equations (36) and (37) that we are led to difference expressions in the transform. Formula (57) will be called the second basic operational property of the Laguerre transform. Let S[F] denote the differential form (38) (xF')'. When the integral T fS[F]} is integrated successively by parts and Ln+ (x) is substituted for Ln(x) - Ln(x) in accordance with property (9) the following result is readily obtained. Theorem 4: Let F(x) denote a function that satisfies. the conditions of Theorem 1. Then T fS[F]f exists and (39) T [S[F]j = -(n+l) f(n+l) (n=O, 1, 2,...). Formula (39) will be called the third basic operational property of the Laguerre transformation T under which the differential operation S[FJ has been replaced by the algebraic operation -(n+l) f(n+l). 12

- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN CHAPTER III OPERATIONAL PROPERTIES 1. The Iterated Operator We note here that the differential form of the fourth order L2[F(x)] is also resolved by the Laguerre transform. If each of the functions L[F(x)] and F(x) satisfy the sufficient conditions for the validity of formula (34) then the transform of the iterated differential form L[L[F]] can be written as (40) T fL2[F(x)] = n2f(n) (n=O, 1, 2,*..). The process can be carried on in a similar fashion for iterations of higher order. 2. Differentiation and Indefinite Integration The operational properties which arise from considering the effect of the Laguerre transform on differentiation and indefinite integration will be given here. Let F(x) be a continuous function whose first order derivative is bounded and integrable on each finite interval in the range x O. Let F(x) be'(eaX), a < 1, as x tends to infinity. One integration by parts of the integral 00 + e-x Ln(x) F'(x) dx, and use of property (10) leads at once to the formula (41) T [F'(x)} = f(k) - F(O). k=O Formula (41) exhibits the image under the Laguerre transform T of the operation of differentiation. 13

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN In looking for a property in connection with indefinite integration we would desire a relation which would give the transform of the integral f F(t) dt fOx in terms of the transform of F(x). Let F(x) be a sectionally continuou.s function over every finite interval in range x > 0, and let G(x) denote the continuous function G(x) = 0 F(t) dt. Then f(n) = ex G'(x) Ln(x) dx = e-x Ln+(x) G(x) dx and it follows from relation (9) for Ln(x) that (42) f(n-1) - f(n) = - e-x G(x)[Ln+l(x) - LA.(x)] dx = -g(n) (n=1,2,3,'); also, 0 f(O) = e'x Gl(x) dx 000 ex G(x) + 0 e-x G(x) dx O 0 = g(O), and _ f(l) = e-x (l-x) G'(x) dx = e-X (l-x) G(x) +f e-XG(x)dx + e-X(1-x) G(x) dx g(O) + g(l). We have used in the above that Lo(x) = 1 and that Ll(x) = l-x. From the difference equation (42) for g(n) and f(n) we have the following conclusion: Theorem 5: If F(x) is sectionally continuous on each finite inter|val over the range O < x <, and O(eaX), a < 1, as x te.nds to infinity, then 14

-ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN (fox (43) Tff F(t) dt = f(n) - f(n-1) (n=l, 2, 3,'), and for n = O, g(O) = f(0) In the above derivation we have used the fact that G(x) is O(eaX), a < 1, as x tends to infinity. Since F(x) has the desired order property it follows by the use of simple properties of the Riemann integral that G(x) does have the aforementioned order property. Solution of the difference equation (42) for f(n) leads to the conclusion: Theorem 6: If G(x) is continuous and Gt (x) sectionally continuous, and if G(O) = 0 and G(x) and GI(x) are ((eax), a < 1, as x tends to infinity, then (44) T {G'(x)} = g(O) + g(l) + g(2) + ~ ~ ~ + g(n) (n=l, 2,..), = g(O) (n = 0). It is interesting now to compare formula (44) with formula (41). We see that we have arrived at the same expression for the transform of a derivative as we had previously with the exception that the term G(0) does not appear here. 3 The Inverse Operator We will consider now the transform of the function L 1[F], where L"1 is the inverse of the differential operator L. Let Y(x) denote the function L-l[F(x)]; then Y(x) is a solution of the differential equation (45 L[Y(x)] = F(x). Suppose that F(x) is a function which is sectionally continuous in every finite interval in the range x > 0 and that 00 (4'6) e-X F(x)' dx =.0; that is that the zero transform of F(x) is zero. It follows from equation (45) that (4) xe'- Y (x) = x F(t) e-t dt 15

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN is then a continuous function of x and has the limit zero as x tends to infinity. Hence Yt (x) is continuous and 9(eaX), a < 1, as x tends to infinity. The second integral can be written (48) Y(x) = f eY I F(t) e-t dt dy + C = L'I[F], Y where C is an arbitrary constant. The function (48) is continuous and can be shown to have the necessary order property as x tends to infinity and hence TfY] exists. According to Theorem 1 and equation (45) then T'L[Y] =.-n T1YJ = f(n); thus (49) TIL-'1[F] = _ f(n) (n=l- 2, 3,."'). n The value of the transform.of L'[F] at n = 0 is given by 00 X (50) TTL-1[F] = A e-X ey. FF(t) et dt dy dx + C. y J The operational property concerning L-1 can be stated as follows: Theorem 7: Let F(x) denote a function which is sectionally continuous in every finite interval in the range x > 0 and let f(O) 0; O also let F(x) satisfy a certain order condition,. Then f(n) exists and for each constant C, (51) T'1 {f(n) = L-I[F(x)] = oX eY oY F(t) e't dt dy + C (n=l,2,..). n O YO 4. Miscellaneous Remarks Theorem 2 gave us the transform of xF'-. We will now, for the sake of completeness, derive the transform of xF. Suppose that F(x) is a function ___1 6_............. 16

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN which satisfies sufficient conditions for its Laguerre transform to exist. The integral 00 eX Ln(x)[xF(x)] dx can be written as e'x [-(n+l) Ln+l(x) + (2n+1 ) Ln(x) n.l(x)] F(x) dx by means of properties (8) and (9) of Ln(x). Hence (52) T{xF - -(n+l) f(n+l) + (2n+l) f(n) - nf(n-1) (n=l, 2, 3.') 00 = f xe-x F(x) dx (n = 0). Subtraction of equation (56) from (52) leads one immediately to the following operational property: (53) T{x(F-F' )} = (n+l) f(n) - nf(n-l) (n=l 2, 3,''.). Equations (52):and (53) are noted to be difference relations. The following theorem follows from the linearity of T: Theorem 8: If T { F(x)3 and T G(x) exist7 then (54) T C1 F(x) + Ca G(x)} CI F'(n) + C2 g(n), where Ci and C2 are constants. When G(x) = 1, then g(n) = 0 for n O and g(~) = 1; according to eqiuation (54) then,. if C is a constant,. (:5) T [F(x) + C} = f(n) when n = 1l 2,', = f(O) + C when n = 0 The convolution property will be discussed in the next chapter. In a later chapter on Sonine transforms I..property ill beiiven which relates Laguerre transforms to Sonine transforms. 17

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN CHAPTT:R IV THE LAGUERRE COiVOLUTION 1. Introdu.ction The convolution property of the transfornation is one that expresses the inverse transform of the product of two transforms in terms of the two object functions withouit use of the inversion formula. We quote now from Churchill and Dolph [4]. "As in the operational calculus based upon Fourier and Laplace transforms the convolution property makes possible a substantial extension of the tables of transforms and it leads to alternate forms, even closed forms of solutions of many boundary value problems." Let F(x) and G(x) be two functions which are sectionally continuous over each finite interval in the range x ~ 0, and Of(eax), a < 1, as x tends to infinity, and let f(n) - L {F(x)b, g(n) = L {G(x)} Then, (56) f(n) g(n) = e-x Ln(x) F(x) dxf e Y Ln(y) G(y) dy = e0 F(x) f e' G(y) Ln(x) Lr(y) dy dx. It will be our aim to write equation (56) in the form (57) f(n3 g(n) = So e't Ln(t) [H(t)] dt. The function H(t) will be the so;-called'onvolution of the functions F(x) and G(x). 18...._...__ 1

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 2. The Addition Property We see in equation (56) that "if we could express the product Ln(x): Ln(y) in terms of a single Laguerre polynomial' we would then have a means of obtaining the form (57). With the aid of the following addition formula from Bateman [1]: (58) 2t r (m+n+l)(-l)n [2(-xy) K]m Tnl(2i x k) Tm(2i y k) c is,1 m e 2 exp [2(-xy)2 kIe im'] To[2ik(x+y) - 4(-xy)2 k cost] dad we will establish the convolution property. Let us now simplify equation (58). Since (59) Tn(x) (. Ln(x) P(m+n+l) we can write relation (58) in terms of Laguerre polynomials. We will also make the following substitution: 2ixk = xT 2iyk = y". Expression (58) now becomes (60) 2'i Ln(x) Ln(Y) =.2 exp (1X ei() LCn(x+y - cos'y ) do, O where we have dropped the primes. We now assert that the imaginary part of the integral in expression (60) is zero. Since (61) eit cos+ i sing the imaginary part of equation (60) can be written as (62) i e[f crsin (xy sin s) Ln(xty - 2 4 cost) dr'+.;/2e x sin (4xy sing) Ln(X+y - 2 qf cost) d ]. 19

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN In the second integral in the brackets make the following substitution: let 2.i - = -. Then the second integral becomes -f eCs n sn sin () Ln(x+y -2x cos 0) do, and the imaginary part of equation (60) is zero. We now have (63) Ln(x) Ln(y) - 0 ex' cos(Fxy sing) Ln(x+y -2 xy cos) d. The form (63) follows by writing the real part of equation (60) in two parts and letting = 2x -' in the integral with limits from X to 2n, Equation (63) is the final form of the addition property as we shall use it in obtaining the convolution property of the Laguerre transforms. Since e C Cos (4xy sing) din = ia and Lo(x) = 1 and Ln(x) = l-x -we see that the above property will check for n.= O and n = 1. When n.= 0 we have 1 1 [>xP e C~Scos(.E sing) dt.l When n.= 1 we have (N-x) (lz-y) e A oScos( sin 2 4- cosbl de 1K $ &xycos'cos (qf sin de r l (x+y)'\ ~ cos( m x+),, eJ COS~cos(4xy sing) dl +2 xy cSY l(cos/x'Y(4 sine) cosr'd(. This expression leads to the identity (l-x)(Z-fy) = 1 - (x+y) + xy. -.20.

ENGINEERING RESEARCH- INSTITUTE ~ UNIVERSITY OF MICHIGAN 3. The Convolution Property In light of equation (57) we will now write the product (63) in a form where the Laguerre polynomial will have an argument of a single variable. Equation (63) will be transformed into x+y+2 /Txy.(x+y-t) co-si[4xy-(x+y+t)2] (64)'tLn(x) Ln(y) =. x e.......-. ).-.}.Ln(t) dt U x~y~~24xy [4xy-(x+y..-t) dt by' letting t = x+y -2 4J -cosy. The product (56) now takes the following form: (65) itf(n) g(n) = fJ J Y " e t Ln(t) F(x) G(y) H(xyt) dt dy dx,,..dx+y-2 4xy where -2(x+y-t) [e14xy- (x+y-t ),.2] H(x,y,t) = I.. [ 4xy - (x.yt. 2:.. We will now proceed to interchange the order of integration in equation (65) since we are aiming for a form similar to equation (57). We see from Figure 1 that the interchange of the order of integration with respect to y and t affects the inner two integrals as follows:. rx++2 Y x e.t Ln(t) F(x) G(y) H(x. y,t) dt dy = JO Jx~ty-2 N dx+t 2 Jxt o x+t24' e"t Ln(t) F(x) G(y) H(xy,t) dy dt. ffx+t.2 4'. ([,X), // / / / Figure 1.......- _ _ 21

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN We can complete the picture now by interchanging the order of integration with respect to x and to The product (57) can now be written in the following form: (66) if(n) g(n) e-t Ln(t) Jx+t2 F(x) G(y) H(x,y,t)dy dx] dt ~o~~ O t-2 t'. where H(x yt) e2-(.X+Y t) co 4xy- (x+y-t )]2 [ 4xy-(X+y-t )2 ] 2 The expression in the square brackets in the product (66) is a.function N(t) whose Laguerre transform is. the produ.ct itf(n) g(n). In this sense N(t) is the convolution F(x) *G(x) of the functions F(x) and G(x). The product (66) is not in a convenient form for checking our result. With this end in mlnd we will attempt a simplification in the form of the function N(t). Consider the region of integration for N(t) as shown in Figure-2:. lo, 1,t,/ /'(t,o) X // Figure 2. In the integral rx+t+2 4xt 2 2 ~ ~/c 1~ ~ (67) N(t) F (x+y-t ) x+tco+2 s [4xy-(x )2t G(y)dy dxo OJ xt -2 N/ xt [4xy" (x+y-t)2]i We will make the change of variable (68) 4xy. (x+y-t)2 = 4y 2 4 yT > 0. 22

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Theny (x-y+t) dy = 4y-' dy' It follows from equation (68) that (y-x-t)2 = 4 (tx-yt2), or y-x-t = + 2 ttx-y'2. Hence if y > x + t then y- x - t = 24tx-y, and if y < x + t then y -x -t = -2Jtx-y. It is easily seen that tx - y42 > O. We consider 4tx - 4yr2 = 4tx - 4xy + (y+x-t)2 = (x+y)2 - 2t(x+y)+ t2 - 4xy + 4xt = (yx)2 + t2 2ty +2tx In light of the above we write the inner integral in equation (67) as follows: (69) f X+t e _(x+y-t) cos[{[4xy - (x+y-t)2] G(y) dy + x.t-2 Jt [4xy - (x+y-t)2]2 x+t+2 J~t 1 2'''+t+ 2 "'e2(x+y-t) ~os2-f4xy.. - (x+.y-t)]~ G(y)C d4y. e+(xtyt) cos~[4xy- (x+y-t)2 y In the expression (69) we will now make the change of variable indicated by equation (68). The resulting integral is (70)t y eNJ/ tx-ty 2 G(x+t-2 I4tx-y2) dyt O j~'tx_y,'2 + e StX_2 os y' G(x+t+2 IJtx-y2) dy' Se".f - tx-yt 2

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Combining the two integrals in'. expression (70) we obtain (71) e-X cos Y" [e txY2 G(x+t-2 tx-y2) + ~O Jtx-yt, e- ~tx-y'2 G(x+t+2 y2)T ] dy' We can now write a second form of the Laguerre convolution, namely,. 00.... (72):tf(n) g(n)= (t etX F(x) x cos Y [e'Jo Jo ~~;o G(x+t-2 /tx.-y2) + e' /tx-Y2 G(x+t+2 4tx ]y2 dy dx dt In the above expression we have dropped the primes on the variable y. In the inner integral in equation (72) we will make the following trigonometric substitution: y = t-t sin O, theny dy = 4'Jt coso@ d The inner integral now assumes the form 273) e~J-~ C8 a ~. 4xt Cosos'0 (753) ~ cos (4xt sin @) e G(x+t.-2 4' cos Q) + e.4t cos G G(x+t+2 4xt cos i) dg The substitution @ =. - 9 in the integral f cos (Jxt sin e,e~ t cos Q G(x+t+2 Jt cos @,d. leads to the form (74) ef eos 0 cos (4t sin.) G(x+t-2 4gt cos Q) da for the integral (73)24

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN We can now write a third form for the product (57)- Use of the integral (74) leads to the form (75) vf(n) g(n) = f e't Ln(t) e-x F(x) Jt eos e cos (4 t sin' ). G(x+t-2 Jt cos 9) d x d dt The foregoing can be summarized in the following theorem, Theorem 9: Let F(x) and G(x) be sectionally continuous functions in every finite interval in the range x > 0, and'(eax), a < 1, as x tends to infinity. Then the produ.ct f(n) g(n) of their Laguerre transforms is the transform of the function H1(t); that is, (76) T-lff(n) g(n)} - H(t) where H(t) is given by the following formula: (77) H(t) ='eX F(x) f e oos e cos cos (t sin @) G(x+t-2-tccs@)d..dx, 4. Remarks Let us note what has taken place in the previou.s section. We started by considering the form (78) gf(n) g(n) = e-X F(x) f e'Y G(y) f Xe'cos cos ( 4 sin, Ln(x+y-2 4xy cos @) dQ dy dx. After a moderate amount of manipulation we arrived at the form (7'9) f)(n) g(n) e' Ln(t) + e-X F(x) et cos (Tx t sin @) G(x+t-2 4xt cos () d@ dx dt. We now notice that we have obtained the form (79) from the product (78) by an interchange of role of one of the functions F(x), G(x) and the Laguerre polynomial involved. One might conjecture at this point that given an addition formula of the type 25

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Pn(x) Pn(y) C= F(x,;y,@) Pn(x,y,G) d@, where C is a constant, for the kernel functions one could immediately write a convolution property for a calculus based upon these particular kernel functions. At the present the only two addition formulas known to the author to have been used to arrive at a convolution property are the Laguerre in the present paper and the Legendre by Churchill and Dolph [4]. The aforementioned interchange has occurred in both places. Churchill and Dolph [4] consider the following product: f(n) g(n) = F(cos O) Pn(cos V') sin I. dLJ G(cos x)Pn(cosx)inrdx,' where Pn(x) is the Legendre polynomial of degree n. They use the addition formula Pn(cs k) Pn(cos ) Pn(Cos) d,v O cos c = cos X cos,. + sin X sin p, cos v. to derive their convolution property. If we rewrite the form of the product f(n) g(n) and take advantage of the addition property we can by the above conjecture immediately write the convolution for the calculus of the Legendre transform, We have f(n) g(n) = i Pn(cosv) sinv [f F(cosp.) sin I G(cosp. cosv + sin 11 sin v cos ) d) d3] dvo This is the form given by Churchill and Dolph [4] in expression (9) on page 96. One difficulty at the present time in pursuing this further is the lack of such addition properties in the literature, To give confidence in the work of section 13 and to get acquainted with the convolution property we will consider here a check of the property in a few simple cases, and then suggest a possible check in the general case. Suppose first that both F(x) and G(x) are constant functions, In this case H(t) should be a constant functions, since f(n) g(n) will be different - 26 -

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN from zero only when n=0. When FW)and G(x) are both constant, H(t) takes the following form: (80) H(t) = e eCOB cos (Nft sin 9) M2 dQ dx Jo o = Mf eXf cs9 cos. (4s sin @) da 1dx where F(x) = M, G(x) = M2 and M.i = M = Since fi e t c os cos (J-t sin 9) d. = n we have H(t) = Mig e-x dx = Mi J'0 Hence H(x) is a constant and will have a transform equal to zero for n' 0 and a transform of Mi for n = 0. This result is seen to check with the product rcf(n) g(n) for F(x) and G(x) constant functions. The product would be zero for n. 0 and iM1M2 = "kM for n = 0. As a second example let us consider the case G(x) e 1 and F(x) arbitrary. H(t) again should turn out to be a.constant function. For F(x) and G(x) having the above forms (81) H(t) e-x F(x) e cos 9 cos (N.xt sin 9) d. dx ='tf e-X F(x) dx = irf(0). Hence H(t) is a constant and also is the constant we would hope for since in this case the product tf(n) g(n) has exactly the value tf(O). Consider now the special case when F(x) is arbitrary but G(x) = Lm(x:; Then by property (15) we can write 27

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN (82) f(n) g(n) =.0 e-t Ln(t) Lm(t) dt f e"Y Lm(x) F(x) dx = if m. n = f(m) if n.= m. On the other hand g(n) = 0 when n i m and g(m) = 1. Hence the left-'hand."'siae -of- expressi6 (82) is f(n) when n = m, and 0 when n m Let us assume that F(x) and G(x) are arbitrary functions in the sense that their Laguerre transforms exist and n=0 n=0 converge and IF'(x) < M ea, a < From the product (79) we consider the function G(x+t-2 fxt cos 9) | We write this function.in terms of its series expansion (83) G(x+t-.2 qxt cos 9) 3 [f e-Y G(y) Im(y) d Lm(x+t-2 Jxt eos ),.m=O Substitution of the expression (83) into the product (79) leads to the product (84):tf(n) g(n) = e-t Ln(t) [ e'x F(x) e-Y G(y) Lm(y) dy mO0.fefxt cos. cOS (NIx sin @) Lm(x+y-2 4. cos@) d9 dx]dt.. Here we have interchanged the order of integration with respect to. and summation, This is legitimate since le...+ cos' Ln(x+t-24 cos 9) cos (4t sin @) g(m) et_. t+x 00 which is independent of @, and from the assumgption on' ig(n) I this series of constants converge. n=0 We hatve used the inequality e"2 IL(x) I < 1, (n.O, 1, 21,., x> 0). 28

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN This inequality can be found in Szego [16b] By the addition property (15) we can simplify the expression (84) as follows: (85) fln) g(n) = et L(t) [ ex F(x) eY G(y) L(y) dy m=O Lm-(X) d Lm(t) dt. We now wish to interchange the order of summation and integration with respect to x. This can be done since X X le-x F(x) g(m) Lm(x) I = e'a IF(x) le - ILm(x) | Ig(m) I < MIg(m) I Hence 00 f(n) g(n) = e-t Ln(t) f(m) g(m) L(t) dt. m=O We can interchange the order of integration and summation here since t t I:e 2 Ln(t) e-2 Lm(t) f(m) g(m) I < If(m) g(m) I The product (84) can now be written 00 00 00 (86) f(n) g(n) = j f e-t Ln(t) Lm(t) dt) e-X F(x) Lm(x) dx m=O J e-Y G(y) Lm(y) dy. By the orthogonality property (5) of Ln(x) we see that every term in the expansion is zero except when m = n. Hence (87) f(n) g(n) = e-x F(x) Ln(x) dx e-Y G(y) Ln(y) dy Expression (87) is an identity by definition of the Laguerre transform. The conditions here could be weakened by using the fact that if the series is multiplied by e'X it can be integrated term by term from zero to 29

ENGINEERING:RESEARCH:INSTITUTE ~ UNIVERSITY OF MICHIGAN infinity and it is only necessary to check the resulting series for ordinary convergence. In view of this we- write expression (85) as CC CO C0 f(n) g(n) = f et Ln(t) g(m) f e'x F(x) Lm(x) dx Lm(t) dt m=O 0C 00 = f e-t Ln(t).g(m) f(m) Lm(t) dt ~ m=O Hence had we required t g(m) f(m) to converge instead of n Ig(m) f(m) I we would have had suffc-Sent behavior to integrate the series, The same argument will hold for the remaining interchange of integration with respect to t and summation, The convolution integral can be given a geometric interpretation. Consider Figure 3 in connection with a possible means of obtaining the interpretation. r L _/s \ X Figure 3. Inr the integral (88) H(t) =2.e'-XxF(x) et cos cos (xt sinQ)G(x+t-2 4 cosQ)Q dx let and L2 = r2 + t- 2r4T Cos Q; then (89) H(t) = 2 ff F(r2) iG(L2) e-r2 e cos (4 Y) dA. (Y>O) 30

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN We can write equation (89) as (90) H(t) = 2et f, G(L2) F(r2) e-(L2+r2)cos (A Yf) d (Y > 0) Hence we have the integral over the upper half plane where X, Y are coordinats of the point in question and r and L are distances from the origin and the point (4T0) for fixed t, respectively* Since the finding of the convolution property has been so closely tied up with the addition property for the Laguerre polynomials it seems natural to consider the question concerning the possibility of obtaining the addition property from the convolution formula. Consider the product itf(n) g(n) =f et Lu(t) e.'x F(x) 7eXJt cos. cos (~xt sin @.) G(x+t-2 4 cos @) ag dx dt when F(x) = Lm(x) and G(y) = Lm(y)* The product then becomes (91) tf(n) g(n) f et Ln(t) M st(x) e cos cos (t Lm(X+t-.2 4 cos @) dg dx dt The left-hand side of the epression (91) is 0 if n m and is x if m n. Hence if we write H(x,t) e= e cos o cos (4/. si:r 6) LI(x+t-2 4x cos @) d.. the product (91) Can be written (92) et Ln(t) Lm,(t dt = f e-t Ln(t) K e Lm(x) H(xtl) dx dt We conclude from the expression (92) that nLM(t) = f e&x L'(x) H(x,t) dt. 31

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN If we assume the uniqueness of the Laguerre transform we can immediately conclude that 1 H(xt) = L(x) LE(t), since this form of H(4x) will have the Laguerre transform Lm(t). In light of the above we have obtained the addition property (15) of the Laguerre polynomials.. That is fLm(t) Lm(x) = f er cos(4St sin Q) Lm(x+t-2 a4t cos 0) d@. The polynomial Lm(t) has an expansion, a finite one, and if it canbe shown that the function I eex Lm(x) H(xt) dx has a convergent expansion,thenawe are justified in using the uniqueness property. 32

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN CHAPTER V TRANSFORMS OF PARTICULAR FUNCTIONS 1. Simple Transforms It follows at once from the orthogonality property of the Laguerre polynomials that when F(x) = K, K a constant, then (93) f(n) = 0 (n f O) f(O) = K The orthogonality property also shows that when F(x) = KLm(x), (m=O, 1, 2...), K a constant, then (94) f(n) = 0 (n m) f(m) = K The following integral (95) e(l-t)x Ln(x) dx = )n t) where 0 < t < lleads to the transform (96) T fetxl (l) 1t) (n=O, 1, 2'..). Since (97) f/'xm e-x Ln(x) dx (-1)n ()jmr (m n), = O (m < n) m an integer, we can write the transform of xm. 335

ENG'INEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Since for x > 0 and X any real number > 0, we have the absolutely convergent expansion (98) xx = (-l)n () Ln(x) [-(x + L) n=O we have (99) T IxX = (-1)nr(x+l) () Expressions (95), (97), and (98) can be found in Wigert [19]. 2. Generating Functions and Laguerre Transforms From the uniformly convergent power series 00 (100) l exp ( ) = Ln(x) t (Itl <1) 1-t n n=O it follows that (101) e-x Ln(x) L. exp dx = t (Itl < 1) From the generating function tn (102) et Jo(2 -xt) = Ln(x) n n=0 it. follows that (103) r e-X Ln(x) [et Jo(2 4xt)] dx = tn Jo~f0~~~~~ nt. From Erdelyi [6] we have the generating function (104)- exp (z Io[ = Ln(x) Ln(y) n ( n=O It follows from equation (104.) that (105) f e-X Ln(x) 1. exp (z 1) 0 [2 - d = L'(y) n ( Iz <l 34

"ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 3. Produ-cts of Transforms According to the convolution property (75) we can write some new transforms. From properties (96) and (98) we find (106) TfJ e-x xm/O e~ xt Cos 0 cos (Txt sin ) er(x+t-2 xt cos e)d dx = 0 when (n>m), = ~< mf /.rT (__m (n m) 1-r l~-r/ \Tr and (-l)mn+ (m /Trr)mn) (n<m). 1-r \- m-n From properties (96) and (101) it follows that (17){ e E.L; expr)j J eSt cos O cos x sin O) er(x+t-24 xt cos @) d@ dx - rn (-1) ( r) 1-r l-r = (n r2n, (I < 1)I (1-r)nl This method can be used to write other transforms; however, the integrals will be in many cases difficult to evaluate. - 35

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 4. TABLE OF LAGUERRE TRANSFORMS f(n) (n=O, 1 2,~). F(x) (O < x < oo) 1. 1 if n = m, 0 if n m Lm(x) (m = O, 1, 2, —.) 2. I if n =O, 0 if n ~ O K (constant) ". (ct ( t )n etx (1 < t < 0) 4. 0 if n>m xm (-l)m m' if n = m (-1)n m(mmn) if n < m 5. tn exp (-t (It <1) tn 6. L et Jo (2fxt) 7. Ln(Y) tn exI (t x+) Io ( t (Jti < 1) 8. 0 if n >m et tm e s cos(Ht sin @) _(1)nm. m, rm ( m\ (1wr)m~l \mn when n < m (_)n,r2n..... et [ *Xlose " (1.r)n*l -r cos (4ft sin @) er(x+t'2 xt cos G) dQ dt (IrI< 1) 1 ~ ~~0. (ljf P(x+i)(n, (x 00, A > O a real number).....n...... 6 36

ENGINEERING- RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 5. TABLE OF OPERATIONAL PROPERTIES F(x) f(n) 1. F(x) ex Ln(x) F(x) dx 2. F(x) + C, C a constant f(n) (n=l, 2, 3,'"') f () + C (n=O) 3. F(x) + Lm(x) f(n) (nwm) f(n) + 1 (n=m) 4 FI(x) f(kI) - F(O) k=O 5 p F(t) dt f(n) - f(n-1) (n=1, 2, 3,"') 0Jo f (O) (n=O) 6. xF"(x) + (1-x) F'(x) -nf(n) (n=O, 1, 2,'-.) 7. xF(x) -(n+l) f(n+l) + (2n+l) f(n) -nf(n-1) 8. xFt'(x) -(n+l) f(n+l) + nf(n) 9. x[F(x) - F'(x)] (n+l) f(n) - nf(n-1) 10. [xFt(x)lt -(n+l) f(n+l) 11. e x [xex F':(x)' -2(n+l) f(n+l) + nf (n) 12. e-x F(x) et c (n) g(n) -~O O cos (46 sin 9) G(x+t-2 ~4xt cos @) d@ dx.... -..~~37

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN CHAPTER VI EXAMPLES AND APPLICATIONS 1. Introduction In this chapter we will indicate possible uses of the Laguerre transform. It was for a while thought that the Laplacian operator in paraloidal coordinates would lead to a natural application of the Laguerre transform. Up to this time, however, nothing promising has resulted. 2. The Transform and Laguerre s Equation Consider the differential equation with the parameter X (108) xF" (x) + (l-x) Ft(x) + %F(x) = 0* The Laguerre transform applied to equation (108) gives (109) (x-n) f(n) = O. From equation (109) we see that for AX n f(n) 0 for all n. Hence there is no function F(x) satisfying equation (109) and the conditions under which the first basic operational property is- valid. For A = n the equation has the polynomial solutions known as Laguerre polynomials. The transformation, when applied to the following differential equation: (110) xVI"(x) + (l-x) Vt(x) + XV(x) = F(x) gives rise to v(n) f(n) (n=o, 1, 2, )-]. Awn If' = n we see that f(n) = 0 for all n. This is connected with a theorem from differential equations to the effect that if x = n-, n an integer, then the system (5) can have a solution only if F(x) is orthogonal to the solution 38

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN of the homogeneous case. In this case F(x) would have to be orthogonal to all the Ln(x). The Laguerre polynomials are complete, however, and hence F(x) O. For X f n we can appeal to the expansion theorem to find V(x). Hence V(x) 7 f(n) Ln(X). X-n n;O The third basic operational property offers an example which leads to a known generating function for the Laguerre polynomials. We can consider here that the Laguerre transform can be used to establish that a particular differential equation is a form of Besselts equation of index zero. The Laguerre transform applied to the differential equation (111) xV"(x) + Vt(x) + XV(x) Q O immediately gives (112) -(n+l) v(n+l) = -Xv(n), (n=O, 1, 2,'-*). We note here that if X = n we would obtain v(n+l) =_. v(n) n+l and hence v(n) O for (n=l, 2, 3'). Hence the only possible choice of V(x) would be a constant function. The only possible choice of v(O) = C which would satisfy the equation would be C = O. When X f n we obtain, (113) (n+v) (n=o, 1, 2 v(n) n+l From the difference equation (113) we see that V(n) V 7 u(~)9.... -....39

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Hence appealing to the inversion formula we write V(x) = V(O) y n L(x)., n=O but T eX Jo (a:2) = n n Hence V(xm) = eX Jo (24/X) V(O) If the equation (111) is solved by series method one will obtain Rainvillets Case II [12]. The nonlogarithmic solu.tion will be (115) V(x) =a xn n=O but (-1)n (x)n = Jo (2 ). n=O Hence for a0 = eX and v(0) = 1 we have a previous result obtained from the generating function of the Laguerre polynomials. For ao = eX and v(O) =1 we have. - e-X eX Jo (2 x) dx = 1 0 or' efe'XJo (2 Ax) dx = 1 or f e'X Jo (2 4x)dx = e We note here that if one makes the change of independent variable, in equation (111), 24x = z, equation (111) will reduce to Besselts equation of index zero,......_ _ _....... _ 40

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Application of property (8) in the Table of Operational Properties to. the simple differential equation (116) xFf (x) - F 0 will serve to illustrate the use of another property which involves difference relations in the transforms, The transformed problem becomes (117) -(n+l) f(n+l) + nf(n) f = 0, or f(n+l) = n-1 f(n) n-+l Hence f(0) is arbitrary and for n = 0: f(l) = -f(O) n = l: f(2) = O n > 2: f(n) 0 Hence 00 F(x) =. f(n) Ln(x) n=O = f(O) + f(l) (l-x) = f(O) f(O) (1-x) = f(O) x and the general solution of equationl(116)is an arbitrary constant times x., We note here that the equation xF.(x) + F(x) = O can not be solved by use of the Laguerre transform, Here wewould. obtain f(n) = f(0) for all..n. and f(0) arbitrary. The series f(0) f. Ln(x) does notcge..The solution of this equation is F(x) =.which is not bounded at the origin.. The use of properties which introduce difference relations seem possibly to be of use in considering problems in ordinary differential equati.ons but the wuse of such properties when considering partial differential equations does not appear promising., 41t

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 3. Partial Differential Equations and the Laguerre Transform We will apply the Laguerre transform to the following problem: (119) +2u + Xa.2U. + (l-x) u, = 0 ay2 ax2 ax (120) u(xO) = F(x) limu(x,y) = 0, y 0oo (121) u(oy) | < M!u(xy) I < M eax, a<l, as x + o%, where F(x) is a function such that its Laguerre series converges and 00 In2 f(n)I n=O converges. Let u.(ny) = T {u(x'y)}. The transformed problem becomes (122) d -n i(ny) 0= dy2 (123) U(n,) = f(nn)lim Ui(n,y) 0=.y -~ O0 Here we use the symbol for ordinary rather than partial differentiation since n is involved in the new problem only as a parameter. Differentiation occurs only with respect to y. We have used the conditions (121) already in writing the transformed problem. The general solution of equation (122) is (124).u(n,y) = C1 e4n + C2 e4ny., where C1 and C2 may be functions of n. In obtaining equations (123) we have interchanged the order of taking the limit as y + X and integration with respect to x. If we verify our final result., we need not be concerned with conditions under which these processes may be interchanged. The simplest conditions under which this

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN interchange of order of operations is valid, as well as the one above concerning partial differentiation with respect to y, involve the uniform convergence with respect to y of the Laguerre integrals and the continuity of the integrals with respect to the two variables- x and y. We see from the transformed boundary conditions above that we must take C2 = 0O The first condition above will give C1. Since U(n,O) _ C; we have C1 = f(n), and thus (125) u-(ny) f= (n) e4n. We can appeal to the inversion formula to write (126) u.(xy) = [f(n) eny] Ln(x) n=O Our aim now is to show that the series (126) found above represents a function u(xt) which satisfies all the conditions of the boundary value problem. The above representation of u.(x>) is seen to satisfy the boundary conditions. If y 0 co u.(x, o) f(n) L n(X) n=O which converges to F(x), and im u.(xy) = ] Ln(). y 3c n=O We will for the present assume the interchange of two infinite processes, namely differentiation and summation. We will see under this interchange that u(xyy) satisfies. the differential equation. Consider the following derivatives: u" ='ln.f(n) e4 jY] Ln(x) by2 n=0 43 -

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN au_ t [f (n) es] LA(x) n=O 2 = I [f(n) e ] L(x) n=O Hence the differential equation becomes I [n(f) e4"-Y] Ln(x) + [f((n)e x e'sy](lx)Ln(x)| n=O rn=O n=O or o [f(n) e4][x Ln(x) + (l-x) LA(x)+ n Ln(x)] n-=O But since xLn(x) + (l-x) L (x)+ ni (x:):=0 is Laguerre's differential equation n we see that the function u(x;y) = [f(n) e4y] Ln(x) n=O satisfies the differential equation* In order to justify the above operations we must show uniform convergence of the derived series. We first consider the.series. (127) (127) [n f(n) e ny] Ln(x), n=O This series mu.st converge uniformly with respect to y. We have by assumption on F(x) that for each fixed x > 0, the series (128) =n f(n) Ln(x),_ ~44

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN converges. According -to Abelt"s test, the new series formed by multiplying the terms of a convergent series- by the corresponding members of a bounded sequence of functions of y, such as e4'ny,y whose functions never increase in value with n, converges uniformly with respect to y. Series (127) therefore converges uniformly with respect to y. The terms of (127) are continuous functions of y, hence the function- 2u(xy)/~y2 represented by that series is continuous with respect to y. We next consider the series 00 (129) ZI[f(n) e"ny] Ln (x) n=O and 00 Z [f(n) e y] Ln(x). x'n=O Since ILn(x) I e2 for all n and all x we have for x < xo, where x is some fixed value of x, If(n) L.(x)j < In f(n)j M we have e ae-ny f (n) Lt(x) < a. e "~ and hence the series, (129) converges uniformly with respect to x, 0 < x < xo, This same statement holds., for series (130). Hence the procedure used to show that'u(x;y) Satisfied the differential equation was justified, Let us consider a possible physical interpretation of the preceding problem. Write equation (119) in the form (131) y (e " -y) + (xe aX) =0 Thus we have a problem in steady state temperatures with conductivity K= = e-x in the y direction and Kg = xe-x in the x direction.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN CHAPTER VII soNI.O TRANSFORMS lo Introduction In this chapter we will.atjoduce a generalized Laguerre transform which we will call the Sonine Transform. We will derive a few properties and then show how the Laguerre transform and Sonine transform are connected through a property on transforms of derivatives. 2. Sonine Transforms The sequence of numbers fa(n) defined by the equation (132) f.(n) e-x x. L~(x) F(x) dx (n=O, 1, 2,'. ), where Lj(x) denotes the generalized Laguerre polynomial of degree n, is the Sonine transform of the function F(x). The integral transformation here will be represented by the symbol T tF(x)j. For functions satisfying fairly general conditions,' Uspensky j1<3], on the interval O < x < 0oo the inverse of this transformation is represented by the expansion of F(x) in a series of the generalized Laguerre polynomials 00 (133) F(x) n f(n) La(x) T'1 ifa(n) (O< x < o). (133) r (a+n+l) n=O 3. Properties of Sonine Polynomials The following list of properties of Sonine polynomials will be useful. (134) FO e"x xa La(x) L(x) dx = r(n+l+a) (n = m) = o (n m).

ENGINEERING RESEARCH INSTITUTE *. UNIVERSITY OF MICHIGAN (135) Lo(x) 1 Li(x) 5 a+l-x (136) L(x) (n+a (-x). (137) (nil) L+l(x) - (2na+l-x) La(x) + (n+a) Lal(x) 0. (138) xy"' + (a+l-x)yt + ny = 0 y = La(x). (139) L -(o) =. n, d.La(X) a+l (140) L(E) -~ (x). dx (141) xL+ (x) = (n+a+l) La(x) - (nil) L+ 1(x) (142) La4'(x) = La(x) L al(x) 4. Operational Properties Let R [F] denote the differential form (143) R [F(x)] -2e [Xa+l e X Ft(x)] When the integral TLR[FJ]is integrated successively by parts and -n La(x) is substitu.ted for R[L4(x)] in accordance with the differential equation (138), the following result is obtained. Theorem 10: Let F(x) denote a function that satisfies these donditions: F'(x) is continuous and F"(x) is sectionally continuous over each finite interval in the range x > 0, F(x) and F' (x) are O(eax), a < 1,.as x tends to infinity. Then T{R TF(x)I] exists and (144) TjR [F(x)]P = -n fa(n) (n=) 1, 2,..)t We note here that the basic operational property for the Sonine transform is the same as the first basic operational property for the Laguerre transform. 47

ENGINEERING RESEARCH.INSTITUTE * UNIVERSITY OF MICHIGAN The differential form of the fourth order R2[F(x)] obtained by applying the operator R dx + (a+l-x) d to R[F(x) ] is also resolved by the Sonine transform T F(x). The resolution.can be written at once as (145) TfR2[F]j = -nTjR[F]j = nzfa(n), (n=O, 1. 2,.). The addition property (58) is not appropriate for use in finding a Sonine convolution property since the polynomial under the integral sign will turn out to be a Laguerre polynomial and hence if we multiply two Sonine transforms of the same order together we will find the function which has this product as its nth Laguerre transform. Let us suppose now that F(x) is contin ot's and F' (x) bounded and integrable. We also assume F(x) is O(eaX), a < 1, as x tends to infinity. Integration by parts.of the integral (146) F' "(x) e- x Lan(x) dx will lead to a property of the Sonine transform which involves derivatives. Let Ft(x).x = dv and e-X:xa L(x) = u.. This leads to 00 (147) TFR' (x)} = e-x xa'4(x) F(x) 0 ",.' (n.a) eX xa-.l La (x) F (x) dx'.e-X xa Lan(x) F(x) dx, or (148) T{FFt(x): - fa(n)- (n+a) fa.'l(n), (n=O, 1, 2,..). In obtaining equation (148). we have used properties (140), (141), and (142). Since d/dx Ln(x)'-Ll (x) we have (149) Tn{F'" = -F(O ) + fl (n-.l) (n=l, 2, 3,* ) where fl(n-l) is the n-l Sonine transform with a = 1. When n = 0 T{F"1(x)}. F(q) ffO)

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN BIBLIOGRAPHY 1. Bateman, H. Partial Differential Equations of Mathematical Physics. New York; Dover Publications, 1944* 2, Churchill, R. V., Faurier Series and Bourdary Valu.e Problems. New Yorkt McGraw Hill Book Co., 1941. 3- Churchill, R.. V,, Modern Operational Mathematics in Engineering. New York: McGraw Hill Book Co., 1944. 4. Churchill, R. V. and Dolph, C. L.. "Inverse Transforms of Produ.cts of Legendre Transforms"., Proceedings of the American Mathematical Society, 5, (1954), 93-100. 5. Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. 1, New York: Interscience Publishers, Inc., 1953, 6. Erdelyi, A., Higher Transcendental Functions, vol. 2, New York: McGraw Hill Book Co., 1954. 7. Hille, E., Proceedings Nattl, Acad. Sci., vol. XII (192.6., 261 -265, 348.-' 8. Ince, E. L., Ordinary Differential Equations, London: Longmans Green and Co., 1926. 9. Jahnke, E. and Ende. F., Tables of Higher Functions, Liepzig, 1952. 10. Kapp K., Theor and Application of Infinite Series, New York: Hafner Publishing Co., 1947. 11. NEagns., W. and Oberhettinger, F., Formulas and Theorems for the S.ecial Function's Of Mathematical Physics, New York: Chei's'ea ublishing Co., 1949, -- - -49

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN BIBLIOGRAPHY (cont.) 12. Rainville, E. D., Intermediate Differential Equations, New York: John Wiley and Sons7 1943. 13. Rainville, E. D, "'Certain Generating Functions and Associated Polynomials"t American Mathematical Monthly, 52, (1945), 239-250. 14. Sneddon, I. N.: Fourier Transforms7 New York: McGraw Hill Book Co., 1951. 15. Sommerfeld, A.$ Partial Differential Equations in Physics, New York: Academic Press, Inc., 1949. 16. Szeg6, G., Orthogonal Polynials, Colloquium Publication7 XXIII. American Mathematical Society, 1939. 17. Tranter, J. C., Integral Transforms in Mathematical Physics7 New York: John Wiley and Sons, 1951. 18. Uspenlsky, J. V., "On the Development of Arbitrary Functions in Series:", Annals off Math., 28(2nd series), (1926-27), 593-6199. 19. Wigert, S., "Contribution a la Theori des Polynomes d'Abel-Laguerr, Arkiv V'r Matematik, Astronomi 6ch Fysik, 15, (1921) 22. 50.