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

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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.

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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

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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

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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

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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).

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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

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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)].

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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:~~~~~~~~

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(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

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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.

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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 ~'

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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

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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.

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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

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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

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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

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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.