An Essay Toward a Unified Theory of Special Functions. (AM-18), Volume 18 9781400882373

Table of contents :
PREFACE
TABLE OF CONTENTS
Chapter I. THE OBJECT AND PLAN OF THIS ESSAY
§1. The General Problem
§2. The Method and General Results of this Essay. The F-equation
Chapter II . REDUCTION TO THE F-EQUATION
§3. Reduction to the F-equation
§4. Notation
§5. Solutions of the F-equation Derived from Familiar Functions
§6. Functions which Satisfy also an Ordinary Differential or Difference Equation of the Second Order
§7. The Uniqueness of the Reductions of §3
§8. An Integral Equation which in Some Cases is Equivalent to the Equation (3. 4)
Chapter III. EXISTENCE AND UNIQUENESS THEOREMS
§9. The Existence and Uniqueness of Solutions of a More General Functional Equation
§10. Some Observations on the Existence Theorems
Chapter IV. METHODS OF TREATING SPECIAL FUNCTIONS BASED ON THE UNIQUENESS THEOREM FOR THE CONDITION F (z0, α) = ϕ(α)
§11. Power Series Solutions
§12. Factorial and Newton Series Solutions
§13. Contour Integral Solutions
§14. Taylor’s Theorem. Generating Functions
§15. Certain Definite Integrals
§16. Relations Among Various Solutions of the F-equation
Chapter V. REMARKS ON SOLUTIONS SUCH THAT F(z, α0) = ψ(z)
§17. The Uniqueness of Solutions such that F(z, α0) = ψ(z)
§18. Differential Formulas
§19. Integral Formulas
Chapter VI. CONCLUSIONS AND UNSOLVED PROBLEMS
§20. Historical Note
§21. Some Conclusions from this Essay
§22. Some Unanswered Questions
Appendix I. SPECIAL FUNCTIONS
Appendix II. OPERATORS
Appendix III. EXAMPLES OF EQUATIONS OF TYPE (3.4) NOT REDUCIBLE TO THE F-EQUATION
Bibliography

Citation preview

A n n a l s o f M a t h e m a t i c s Studies

Number 18

An essay toward A UNIFIED THEORY OF SPECIAL FUNCTIONS based upon the functional equation

^

F(z, a) = F(z, a+1)

Sy

c\

TRUESDELL

PRINCETON PRINCETON UNIVERSITY PRESS LONDON: GEOFFREY CUMBERLEGE OXFORD UNIVERSITY PRESS

1948

Copyright 1948

Pr inceton University Press

Photo-Lithoprint Reproduction

EDWARDS AN N

BROTHERS, Lithoprinters

ARBOR,

1948

IN C.

MICHIGAN

In Memoriam H. B a t e man

PREFACE Nature of This E s s a y . This e s say presents and explains one particular branch of analysis, specially developed to provide a new m ethod of d e d ucing and proving formal relations among special functions, a new method w h i ch turns out to be very powerful.

In no sense a n e xpository work, it does not aim

to compile i n f o rmation o n special functions or to outline k n o w n methods of treating them.

The scattered and partial

results obtained b y previous students of differentialdifference equations of the type studied in this essay are summarized in the histo r i c a l note at the end (§20). The essay is w r i t t e n f r o m the point of view of a mathematician.

Readers interested only in practical use

of special functions are directed to Chapter I, §3 , §4,

§5 of Chapter II, C o rollary [9.k] and the remarks f o l l o w ­ ing it in Chapter III, Chapter IV, and Chapter V. Appendices. Material w e l l k n o w n to man y readers or otherwise u n ­ suitable for the body of the essay ha3 b e e n relegated to three a p p e n d i c e s . Enumeration. Formulas are numbered w i t h i n each section.

In refer­

ences to formulas of other sections the number of the sect i o n is prefixed to the number of the formula, e.g., f o rmula (11.6) is the formula numbered (6) in §1 1 . C a p i ­ tal L a t i n letters refer to formulas in Appendix I, e.g., for m u l a (C.3 ) is the formula numbered (3 ) in part C of App e n d i x I. References. References i n square brackets refer to the list at the e'nd of the essay. I

II

PREFACE

Circumstances of C o m p o s i t i o n . I b e g a n the studies leading to thi3 essay through encountering a special case [Truesdell 1]. In A n n Arbor in the s\:jnmer of 1 9^

I proved limited forms of the

existence and uniqueness theorems of §9 and discovered the essential r e duction in §3 .

The m a i n body of results

in §5 ~§8 , §10-§i7 , and §19 I worked out between December *i9bb and September 1 9^5 in leisure hours while a Staff Member at the Rad i a t i o n Laboratory, Massachusetts I n s t i ­ tute of Technology. I n the summer of 19^5 the late Professor B a t e m a n read the manuscript and supplied a number of references, through w h i c h I became acquainted w i t h the previous results of Appell, Bruwier, and Doetsch discussed in §2 0 .

Some of the material in §3~§5> §11 -§l 6

and §1 9 was presented to the A m e r i c a n Nhthematical Society September 1 7 , 19^5 (Bull. Am. Math. Soc. 51 Abstract No. 229)-

(19^5) p. 883,

I n leisure hours while employed at the

Naval Ordnance Laboratory I made extensive revisions and added §1 8 in June 191*6 ; later I put §9~§ 1 0 into their p r e s ­ ent form and presented them to the A mer i c a n Mathematical Society December 2 7 , 19^6 (Bull. Am. Math. Soc. 53 0 9^7)* p. 5 9 , Abstract No. 5 7 ). Finally a summary of some of the contents of this essay was published in the Procedings of the National Academy of Sciences [Truesdell 2 ]. Acknowledgement. I w ish to express my obligation to the late Professor B a t e m a n and to thank Professors B. F r i e d m a n and W. Hurewicz and Dr. Max M. Munk for reading and criticizing p o r ­ tions of this essay. Miss Charlotte Brudno and Mrs. Peggy Matheny have kin d l y assisted in the preparation of the final manuscript. C.A.T. Naval Ordnance Laboratory, W h i t e Oak, Silver Spring, Md. and Unive r s i t y of Maryland, College Park, Maryland. M ay 1, 1 9^ 7 .

I ll

TABLE OF CONTENTS Page Chapter I .I . THE OBJECT THE OBJECT AND PLAN AND POF 1ANTHIS OP THIS ESSAY ESSAY § 1 . The G eneral Problem .............................................. § 2.

The Method and G eneral R e s u lts o f t h i s E ssa y . The F -e q u a tio n . . x......................

C hapter I I . REDUCTION TO THE F-EQUATION § 3 . R ed u ctio n to th e F - e q u a t i o n ......................... § §

4 k. .

§

6.

§ §

N o t a t i o n ......................... N o t a t i o n ................................................................... 5. S o lu tio n s o f th e F -e q u a tio n D erived from F a m ilia r F u n c tio n s ............................... F u n ctio n s which S a t i s f y a ls o an O rdinary D i f f e r e n t i a l o r D iffe r e n c e E q u atio n o f th e Second O r d e r .................. 7. The Uniqueness o f th e R ed uctions o f §3 . 8. An I n t e g r a l E q u atio n which in Some Cases i s E q u iv a le n t to th e Equa­ t i o n ( 3 - 4 ) ..........................................................

C hapter I I I . § 9.

§10. C hapter IV.

EXISTENCE AND UNIQUENESS THEOREMS

The E x is te n c e and U niqueness o f S o lu ­ tio n s o f a More G eneral F u n c tio n a l E q u a t i o n .............................................................. Some O b servatio n s on th e E x is te n c e T h e o r e m s .............................................................. METHODS OF TREATING SPECIAL FUNCTIONS BASED ON THE UNIQUENESS THEOREM FOR THE CONDITION F (z q , a) = 4>(ot)

§11 .

Power S e r ie s S o l u t i o n s .....................................

§12.

F a c t o r ia l and Newton S e r ie s S o lu tio n s .

11 1 10 U

14

16 16

17 23 28

35 39 39 50

55 55

.

69

§13. Contour I n t e g r a l S o lu tio n s ............................ § 14 . T a y lo r ’ s Theorem. G en eratin g F u n ctio n s.

74

§15. §16.

C e r ta in D e fin it e I n t e g r a l s ............................ 105 R e la tio n s Among V a rio u s S o lu tio n s of th e F -e q u a tio n . . . . ................................ 116

C hapter V .

REMARKS ON SOLUTIONS SUCH THAT F ( z , c*o ) = * ( z ) The Uniqueness o f S o lu tio n s such th a t F ( z , n) d n ^ , g(y, n) 4 f n (y)> n d [ h ( y )] n n Is a positive integer? Transformationa:

(1 7 )

e^F^b

- a; b; - y) =

(1 8 )

e^F^a

+ p

5; y 2 ) =• ^ ( a ;

(a; b; y),

2a; 2y),

a (19) (20)

r(a + 1 )y

2 Ja (2f D

= qF 1 (a + 1; - y),

(1 - y ) " a F(a, c - b; c; ^ - ) = F(a, b; c; y).

To formulate a n inquiry into a general class of t r a n s f o r ­ mations w h i c h includes all the formulas (1 7 ) through (2 0 ) is rather difficult, because it is hard to find a clear distinction b e t w e e n them and a n entirely arbitrary f u n c ­ tional transformation. Tentatively w e suggest

§1 . Que s t i o n h. q(y)

T H E GEN E R A L PROBLEM

5

Do there exist functions

h(n, y)

and

such that

(D)

h(n, y)f (q(y)) =

wh ere the

an

5 a ny n , 1>=0

c an be exhibited explicitly!

The word

"explicitly” leaves considerable vagueness in this question. Contour I n t e g r a l s : (Jy)a (2 1 )

Ja (y) -

2Vi

v2 exp(w " W )dw'

(o+) 1 w a

( 2 2 ) J.(y) a

(2 3 )

( 2h )

(25)

r ( j - « ) ( h ) (i+,ri-) P a-£------------:---i (w -1 ) cos 27ri r ( p a 1

(^

)

O

PA (y) - — ------------------a 2 2iri A

Q

S

w y dw,

_Q —1

(W -1 )

(w-y) a

P (y) - ~ $ [y + V y 2 - 1 cos w ] a+1 dw,

l £ a ) ( y ) = ES ±§ h 1 ( ° f V

Supposing ^ n (x ) natural to ask: Question 5 .

‘ ^ ) nw ' a "1 ew dw.

belongs to pur class of functions, it i C a n we find a contour integral for

f n (x)f Definite I n t e g r a l s :

dw,

I.

6

THE OBJECT A N D PIAN OP THIS ESS A Y

(2 7 )

7) o

_ -at *r / ^ u d - l , , b cr(c+d) -m/C+d c+d+1 _ , . b 2 v e J (bt)t dt = - ■ — *----1— F (-^-,— — ;c + 1;-- -), c 2ca c+dr(c+l) 2 2 c

(2 8 )

7o»J

p ?

r ( c+A^oc

. " b n ,d-i Jt (at)dt /-i . '.3 j . — 2 , + d . «,i . ,F .T(? / C ,c+i o c T(C+1)2c+1b c+d 1 1 2

(2 5 )

I e-b t tai ^ a ) (t)dt = o ml Question 6.

T e"btCta f ( t ) d t o n

be evaluated in general?

(3 0 )

What other integrals involving

is it natural to set up? Integro-difference R e l a t i o n s : — y 2 Jb + a (2v7) =

a-b-l (3 1 ) y 2 (3 2 )

.

Can the integral

(E)

fn (y)

b

p

a p), * kb2

y 1 a J w 2 (y-w)b_1 Ja (2 v^T)dw,

^ =

fTbTTT

L o

_la 2 Ja(2i^t)dt,

1 ta d - t ) b_1L(a ) (ty)dt .

Question 7 . For a giv e n function fn(y) ca n s u i t ­ able functions g(w, a, n) and h(w, y) and numbers b and c be found such that (F)

fn+a(y) =

B(w'

n) f n (h(w’y ))dy ?

§1 .

T H E G E N E R A L P RO B L E M

7

Miscellaneous R e l a t i o n s :

(33)

e“ yy 2

L^,a ) (y) - r ( b 1+ 1 ) | e -tt

2 Ja (2i/ty)dt

- vTrfcry I ^ X ^ y ) ^ , ( 35)

Pa (y) = r ( a -1b+1 ) J e ' t yJb(tvT ^2 )tadt. o

Q u e s t i o n 8 . Is there a general w a y to find a r e l a ­ t i o n e x pressing one individual mem ber of our class of functions in terms of a second individual member? W e shall t ry to answ e r Questions 1 to 8 . In other words, The a i m of t h i 3 e s say is to -provide a general theo r y w h i c h motivates, discovers, and coordinates such s eemingly u n c o nnected relations among familiar special functions as the formulas (1 ) through (3 5 )* I emphasize the words "motivate" and "discover." It is no great task to c o n s t m e t ad hoc rigorous proofs and to find the range of val i d i t y of a ny of the formulas we have just listed, once one sees t h e m set up; their d e r i v a ­ tions f o r m typical exercises for the student in English text books. No text book, however, suggests as a pro b l e m "Find a formula w h i c h gives Laguerre polynomials in terms of Bes s e l functions," because the student would have no idea w h e r e to start unless he h a d the advantage of having s e e n such a f o rmula before. The discoverers of these formulas hav e used the i r i n t u i t i o n combined o f t e n w i t h b rilliant artifices and have discovered t h e m singly, g iving us slight i n d i c a t i o n how to find others like them. I n this e s say consequently, I a i m to give rational methods of d i s c o v e r y ; if l i m i t a t i o n of space, inattention, or ignorance has not always p e r m itted me to satisfy the reader’ s standards of rigor, still I believe that it is

a

I.

THE OBJECT A N D PLAN OF THIS E S S A Y

possible to supply the missing details and to. correct any erroneous o n e s . I emphasize also the word "coordinate." It is my aim to bring order into a part of the collection of k n o w n relations concerning special functions by showing that they are simple special cases of about a dozen general formulas and by adding to their number some of the missing analogues w h ic h do not seem to have been discovered thus far. F i nally I emphasize the words "familiar special functions." This essay does not consider a n e q uation for its o w n sake and the n seek functions satisfying it whi c h m ay serve as examples, but rather it observes and inv e s ­ tigates a hitherto largely neglected property common to the transcendents occurring most frequently in m a t h e ­ matical physics and shows that it is this very property wh i c h entails as necessary consequences such relations as the formulas (1 L ooking at tions involving eight questions

) to (3 5 )« a typical collection of 35 formal r e l a ­ familiar special functions, we have listed w hich come to mind immediately, and we

have seen that in order to give them any real meaning we must refer them to the theory of some suitable class of functions. Our first problem, then, is to find this class. W e wish to include Bessel, Legendre, Laguerre, and Hermite functions, so let us examine the major formal properties these functions have in common: (1 ) They satisfy ordinary linear differential equations of the second order; (2 ) they satisfy ordinary linear difference equations of the second order; (3 ) with suitable weight functions they f o r m complete sets of orthogonal functions over suitable intervals; (4) they satisfy linear d ifferential-difference relations. The first three of these properties after very long and thorough investiga­ tions by numerous excellent mathematicians have yielded but slight clews to the discovery of such relations as

§1 .

T H E G E NERAL P ROBLEM

the formulas (1 ) to (35)*

9

Our confidence in the essential

bea u t y and p e r f e c t i o n of classical analysis would be sha k e n if in fact these formulas were, so to speak, random effusions of the Divine Mathematician, disjoint and chaotic, so w e are d r i v e n to conclude that they are c o n ­ sequences of some of the differential recurrence rela-

tions (36)

^ Js (y) = |

(37)

Ja (y }

(38)

( 1 -y2 )iy

(*0)

=

Pb (y)

pj(y) =

Ja (y)'

" Pj+i(y)>,

(a+b)pj_1 (y)

^

y

(*3)

y

(U) y

^

L^b ) (y) L^b ) (y)

L ^ b ) (y)

-

ayPb (y),

Pa+1(y) • by Pa(y^

(a-b+1 )(a+b)Vl"-y2 Pb " 1 (y)

^ L ^ ( y )

(*2)

( **-7)

Ja + i (y ) >

"f

(1-y2) ^ pj(y) = -

(itl) (1 -y2 %

(*6)

Ja - i (y)

'

Pa (y) = (a-b+ U ( y P b (y)

(39)

(*5)

"

Ja (y)

=X

+ by P b (y)

= L ^ b ) (y) - L(y),

= -

(a+b)La b _ 1 5 (y) - bL^b ) (y), a L^b ) (y) - (a+b)L^b ] ( y ), a

+

1

'

(a+b+1-y)L^b ^ ( y ),

^ H a (y) = 2yHa (y) - H & + 1 (y), ^

Ha (y) = 2aHa _,(y).

Ac c o r d i n g l y w e shall study functional equations of this type.

I. §2.

T H E OBJECT A N D PLAN OF THIS ESSAY

THE METHOD A N D GENERAL RESULTS OF THIS ESSAY. THE

F-EQUATION

W e are going to study functions a functional e quation of the type ) 3y where

^ = A(y, cx)

B(y, 01) +

^

and

f(y, cx)

B(y, cx)

satisfying

ot+1

are given functions, and

does not vanish identically.

W e shall show that

by a proper transformation, which we shall exhibit e x p l i ­ citly, this e quation may be reduced to the form (2)

g(y> «■) = C(y, oc) g(y, ot+1 ),

and that w h e n C(y, oc) = Y(y) A( -1 , and conclude immediately that z °^2 Joc (2t/z ") is a n integral f u n ction of sion

z

with the power series e x p a n ­

§2.

T HE

F-EQUA T ION

/0 * co / \n n -oc/2 J (pvc-\ _ (1 .33)> and (1.35) are all special cases of a second integral formula. W e shall men t i o n other definite integrals who s e integrands contain solutions of

ANDOBJECT PIAN AND OP THIS 12I . THE OBJECT I. THE PLANESSAY OF THIS ESSAY the F-equation, and sketch a method for building up inte­ gral formulas in increasing complexity. Hence we shall answer Question 6 adequately (§15)* We shall show that the formulas (1.30), (1.31), and (1 .3 2 ) are all special cases of the same integro-difference relation. Hence we shall answer Question 7 (§18). We shall give a straightforward general method of attacking such problems as "Find a formula giving the Laguerre function L^a ^(y) in terms of the Bessel func­ tion Jc (y )M which will lead automatically to the formula (1.33), and will serve equally well to find an inverse for it. The method will with equal ease discover the formula (1 .3 M and an inverse for it, It does not seem to apply easily to the formula (1.35)> which, however we may discover very simply by means of another general formula satisfied by a class of solutions of the Fequation. Hence we may go far to answer Question 8 (§1 6 ). We shall make some headway in coordinating formal relations satisfied by familiar special functions by showing that a great many of them are included as special cases of about a dozen general theorems concerning solu­ tions of the F-equation. For example, we shall see that the proper analogue for the Bessel functions of the Legendre function formula (1 .2 ) is not the formula (1 .1 ) but the formula (1. 5). While answering these questions we shall find other topics of interest to be investigated, and in the end we shall have a fair insight into some of the formal proper­ ties of familiar special functions. It is not my aim to produce a long list of new rela­ tions satisfied by various special functions, but rather to render trivial the discovery and proof of a large class of these formulas. As illustrations I have applied the results of this investigation to deduce sometimes a formula I had already known, and sometimes one previously unknown to me which I had discovered (quite possibly re­ discovered) by the methods here presented..

§2. I

THE

P - E QUATION

13

hope that I shall be f o rgiven for listing a rather

large number of examples.

The methods we discuss in this

essay are so easy and elementary that without inductive evidence to the contrary I fear the reader might doubt they could be good for anything.

Chapter II REDUCTION TO THE §3.

REDUCTION TO THE

Suppose the f u n ction tional equation (1 )

sy

f(y,

F-EQUATION

oc)

satisfies the f u n c ­

= A (y> °o f (y> °o + B (y> cx+1 )•

Let the f unction ti o n

g(y, cx)

be defined by the transf o r m a ­

y g(y,