The Early History of the Hypergeometric Function JACQUES D U T K A

Communicated by M. KLINE

1. Introduction

The hypergeometric function, represented when Ix l < 1 by the series

a. b +a(a -4:- 1) b(b + 1) F(a, b; c; x) -= 1 -+- l~.c x 2! c(c + 1) x 2 a(a -[- 1) (a + 2) b(b + 1) (b + 2) X2 ~3! c(c + 1) (c -t- 2) -+- ....

(1)

and the closely related gamma function are among the most irnportant in the class of special functions in mathematical analysis, and also arise frequently in applications. Yet most books and monographs which treat the hypergeometric function say little more about its early history than to make a brief reference to some results of EULER and then turn to the fundamental developments of GAUSS,KUMMER,RIEMANN,and others. This does scant justice to the contributions of numerous other mathematicians, ranging in time from the mid-seventeenth to the early part of the nineteenth centuries, which provided many important ideas and motivations for the more sophisticated developments of their successors. It is the purpose of this paper to call attention to some of the earlier contributions and of the motivations of the mathematicians who made them.

2. The work of John Wallis and his successors

The latter part of WALLIS'S Arithmetica Infinitorum contains a number of results and concepts which proved fruitful when developed by WALLIS and his successors. Notable is WALLIS'S concept of "interpolation." Now methods for determining the values of intermediate values in tables or the extension of such tables from adjacent values, already obtained, were known since ancient times. But WALLIS'S interpolation amounted to extending a number sequence {an} defined for n = 0, 1,2, ..., to non-integral values of n by a "law of continuity."

16

J. DUTKA

E.g. WALLIS [1, Prop. 133] considers the sequence

1,

1

1

1

6'

30'

140 . . . .

(1)

generated from 1

f (x -- x2)" d x ~ n!/(n + 1) (n -k 2 ) . . . (2n + 1),

n = 0, 1, 2, 3, ...,

0

and finds that the value corresponding to n = ½, which in (1) would come between 1 and 4, is, on geometrical grounds, a/8, the area of a semicircle of unit diameter. Similarly, in Proposition 161, he obtains the sequence 1,

5 6'

31 30'

209 140 . . . .

(2)

generated from ,

f

(n -k m + 1),

(x + X 2 ) n d x =

0

n = 0, 1, 2, 3, ...,

m=O

and seeks the value corresponding to n = ½, between 1 and :5 in (2), the area of a hyperbolic segment. He is unable to evaluate this, but by methods of the calculus, developed several decades later, it is [3 ]/8- -- In (3 + 1/8-)]/16. Another sequence which plays an indispensable role in obtaining his culminating result, a product representation of 4/~, is 1,

3 2'

15 8 '

105 48 . . . .

(3)

generated from f ( p , q) = 1 / j ( l - - x } ) q d x

p=

1, 2, 3 , . . . ,

/

-

(pp !+q ! q)! '

(4)

q=0,1,2 ....

WALLIS'Sprincipal objective was to find an analytical expression for the reciprocal of the area of the quadrant of the unit circle, f(½, ½), which he denotes by []. In contrast to the geometric progression a, at, ar 2 . . . . in which each term, after the first, is obtained by multiplying the preceding term by a constant ratio, WALLIS [1, Scholium to Prop. 190] and [2, pp. 315-316] introduced another type of progression, to which he later gave the name "hypergeometric," in which the successive multipliers are unequal. (E.g., the factorial sequence, 1, 2, 6, 24 . . . . . is a hypergeometrical progression.) Now since in a geometric progression each term, after the first, is the geometric mean of its predecessor and successor, he introduces an analogous mean. E.g., in (3) he writes ~ = ~ : 1 I ~ and has [ ] ---~ : 1 ]3. He ultimately evaluates this interpolated value in the form of the remarkable infinite product 4

[] =--Y =

3

3

5

5

7

7

"

(5)

History of the Hypergeometric Function

17

Corresponding to the sequence (3), he also obtains a sequence 4 3'

1,

28 15'

288 105 . . . .

(6)

generated from f (1 + x 2 ) n d x =

(2mq-1),

0

n=0,1,2 ....

m=0

and seeks the value corresponding to n = ½. (He does not evaluate the area of this hyperbolic segment which is [1/2" + In (1 -}- ]/-2)]/2.) Although WALLIS'Smethods were largely supplanted by those of the calculus within a few decades, his ideas had an important influence on his successors. In his famous second letter to OLDENBURG (intended for LEmNIZ) of 24 October 1676, NEWTON [I, 130ft.] tells how he applied WALLIS'Smethod of interpolation, at the beginning of his mathematical studies, to develop the binomial theorem. It is also in this letter that he discusses the quadrature of the class of curves whose ordinate is dz°(e + f z ~ ) ~ where d, e, f a r e constants and 0, ~/, 2 are exponents. (This includes the quadratures previously considered by WALLIS, described above, and may have been intended as a further development of WALLIS'Sprocedure.) NEWTON sets (0 + 1)ill = r,

2 + r = s,

d" (e + fz~)Z+lfilf = Q

and

r r / - - ~7 = z~

and obtains for the area of the curve the equivalent of --~-Q.F

1, 1--•/;

l--s;

--

(7)

in the notation of (1.1). (This can readily be derived by a reduction formula obtained from g z~

eY~

s f z ° (e -?fzn) a az = - ~ f ( e --}-fzn) ~+' - ~l--f f z°~ (e + f z ~ ) ~ az

and then by replacing 0 by 0 - ~/, 0 - 2~/, 0 3~/. . . . successively. From (7), one can verify NEWTON'S statement that the series is infinite when r is a fraction or a negative number, but is finite with r terms when r is a positive integer.) The influence of WALLISon the work of STIRLINGand EULER will be discussed below.

3. The work of St~ling

JAMES STIRLING (1692-1770) was a disciple of NEWTON; the title of his principal work [STmHNG, 1] is taken from a tract of NEWTON'S. Despite its title, it is primarily concerned with the summation and interpolation of series. (A useful summary of this work was given by C. TWEEDIE [1].)

18

J. DtrrKA

STIRLING [1, pp. 16, 113-114] mentions a series of NEWTON given in the latter's De Quadratura [NEWTON, 2, Prop. V, Th. III] to the effect that x0 fz°-1(e+fz~)Zdz=-~e(e+fx~)Z+~F o

(

0 0 1, - - ~ - + 2 + 1; - - + 1 ; ~7

~) --

, (1)

(for ] --fxn/e] < 1) which is here written in modern notation. (This follows from a reduction formula

f

zo-1(e + fz~) ~ dz

z°(e + fz~) a+l 0 + 2~7 -k ~1 f Oe -0 e- f z°+~-l(e + fzn)~ dz

by applying the identity to the integral on the right, etc.) STIRLING obtained the equivalent of

/ o

eZx ° (~ z°-l(e + f z ' ) ;~dz = - - . F 0

'

0 _~) -- 2; - - 4- 1 ; -~

'

(2)

(for [--fx~/e] < 1) by expanding the binomial factor on the left and integrating term by term, and of

f zO_l( e +fz~);.d z = xO (e -kfx~) ~". F 0

o

1, - - 2 ;

0

~+

1;

fx ~ e +fx']

(3)

for lfx'/(e +fx~)] < I). (This follows from a reduction formula

f z °- I(e + fz'f" dz = z ° (e + 0f z ' ) ~

2f0 f z°+~-l(e +fzn)a-1 dz

by applying the identity to the integral on the right, etc.) STmLmG points out that his series (3) terminates if 2 is a positive integer and that the series from (1) is or is not preferable to (3) according as e a n d f have unlike or like signs respectively. There are many places in his work in which STtRLING shows the influence of WALLIS[1] including his interest (Prop. XXI, Ex. II) in the problem of interpolating in the sequence if(n)} where f(n) = n!, n = 0, 1, 2 . . . . . (About the same time that STIRLING was writing his book, D. BERNOULLI, C. GOLDBACHand L. EULER were also investigating this problem independently.) By a remarkable numerical analysis, STmLING confirmed a result which would now be written in the form ~(½) = ( - ~ ) ~ = ¢~. Noting that the terms of the sequence are generated by the equivalent of f(n + 1) = nf(n), STIRLING first tabulates log 5!, log 6!, ..., log 17!. Then the successive even differences from the second to the tenth order are tabulated. He then seeks to interpolate a term which would correspond to log((10.5)!). He applies a central difference interpolation formula (p. 111), afterwards attributed to the German astronomer F. W. BESSEL, and finds that (10.5)! -- 11899423.08. By a backward recursion in which, in effect, he divides by 10.5, 9.5, 8.5, ..., 1.5 successively, he finds that (½)! -- 0.8862269251. He states that the square of this term is equal to the area of a circle of unit diameter and that the term whick

History of the Hypergeometric Function

19

immediately precedes this is l/~ (that is, ( - - 5 ) ! = I/m) • He does not explain how he recognized that (5)l = 1/~/2 from its numerical value but mentions the sequence {(n !)2} and states that sequences of this type may be interpolated without the use of logarithms as shown subsequently. Following this, STmLING several times considers interpolating in the sequence {(22)/22n } n = 0 , 1 , 2

.....

and i n t h e reciprocal sequence {22n/(22)},

and

develops various series for the squares of these sequences. (The question raised at the end of the preceding paragraph is not answered directly. But it is an immediate consequence of WALLIS'S result (2.1), by interpolating between the first two terms of this sequence. C f also WALLIS'S Prop. 165.) STmLIN6'S series in the form given by J. BINET [1] are 2 z"

=~n'F(½,5;

n+l;

=-~-(2n + 1) F(5,

1)

5; n -l- 1; 1), (4)

2~

--zl(2n + 1) F(5, 5; n q--~; 1

=--F(5,

--5, n + 5 ;

~n

1)

1).

(The first and third were actually given by STIRLING;the second and fourth are BINET'S corrections of those originally given by STIRLIr~G.) The results follow directly from GAUSS' relation F(a, b; c; 1) = 1"(c) I'(c -- a -- b)/[F(c -- a) (c -- b)]

if c -- a -- b > 0.)

A remarkable result is embodied in STIRLING'S Prop. XXIV in which he introduces the equivalent of the integral i

B(r q- z , p -- r) --= f xr+Z-l(1 -- x) p - r - I dx

(5)

0

and uses it to interpolate in the sequence a,

r

--a, p

r(r + 1)

p(p -1- 1)

a,....

(6)

(The term "beta function" for (5) is due to J. B~NET.) STmLINO shows the equivalent of B(r -f- n,p)

r(r ÷ I ) . . . (r q- n -- 1)

B(r,p)

( p + r ) ( p q- r + 1)... (p q- r + n -- 1)"

(7)

20

J. DUTKA (In particular, he uses the foregoing to interpolate a term corresponding to

n=½

in the sequence {(Znn)/22"}, n = 0 , 1 , 2 1

. . . . i n t h e form

1

f (1 -- x ) - : d x o 1

2

1

7~

.f (x - x 2)- ~ d x 0

C f the numerical result obtained by STIRLING above.) STIRLIN&S best known result, the approximation for log n!, is given in his Prop. XXVIII (pp. 135-139).

4. The work of Euler

Researches on the determination of general terms of sequences and, in particular, on interpolation in the sequence of factorials were made in 1728 and 1729 by CHRISTIAN GOLDBACHand DANIEL BERNOULLI,but it was the then twentytwo year old friend and compatriot of the latter, LEOI~HARD EULER, who made a decisive step forward in a remarkable memoir submitted to St. Petersburg Academy in November 1729. In this memoir [EULER, 1] (and in an earlier letter to GOLDBACH), EULER gives the formula 1 - 2 n 21~n.3 n 3 I - n . 4 " 4 I - " . 5 " n! = - --... l+n 2+n 3+n 4+n

(1)

without proof, but verifies it for n -----0, 1, 2, 3. In particular, for n = ½, EULER finds from WALLIS'S formula (2.5) (k)l =

V ~ "4 4 " 6 -3 5 5

6"8 7 7

8"10 V4 • ". . . .

(2)

EULER'S own proof of (1) was not published until more than sixty years later in [EULER, 2], and does not appear to be well known. (It is not mentioned in the modern expositions by P. J. DAvis [1] and C.J. SCRmA [1].) A sketch is given here. In EULER'S notation, let A : i = 1 • 2 • 3 .., i. Then for a fixed g and a positive integer n

(i + o~)"

1 • 2 . . . i(i + e~)"

1 =lim = l i m .. i-+o~(i + 1) (i + 2 ) . . . (i + n) i+~o (t + 1) (i + 2 ) . . . (i + n) A : i

•,

= lim i~(n

1 • 2 . . . i(i + o0" + 1)(n + 2 ) . . . ( n + i ) A :n'

whence it follows that 1 • 2 . . . i(1 + ~)n A : n ~ lim A (n, i) = lim-, . " , , i - + o o t n t 1)(n + 2) ,.. (n + i)

History of the Hypergeometric Function

21

By a "continuity" argument (like that of WALLIS), n is now not restricted to being an integer and one has 1 • 2... n!

i(1 H c¢)"

(3)

= i ~ ' ~ ( n + 1) (n H 2) ... (n + i)"

(EULER [2] gives this formula with o¢ = 1. Essentially this formula was later used by GAUSS [1, ¶ 20], with o~ -----0 as the definition of I'(n + 1) ----n!.) Now from the definition of (An ,i) (1 + ~ ) " (n + 1) '

A(n, I)

A(n,m+l) A(n, m)

mill ( m + l +o~)" n + m + 1 (m q- Lx)"

Thus, finally, on setting o~ = 1, and m = 1, 2, 3, ..., successively, one gets 1 • 2" (nil1)

A : n - - - -

2.3" 3 • 4" ( n i l 2 ) 2" ( n q - 3 ) 3 " " "

that is, (1). To find a more convenient expression for n!, EULER [1] (like SXmLING) intro1

duces WALUS'S integral, f xe(1 -- x") dx, where n is a non-negative integer and o e is arbitrary (but =~ --1, --2 . . . . , --n). By expanding the binomial factor in the integrand EULER found that 1

1.2...n

f xe(1 -- x ) " d x -- (e H 1)(e H 2) 0

"*"

(e + n q- 1)"

On first setting e = f i g and then f ---- 1 in this expression, EULER then shows the equivalent of 1

n! = lira ~-4--i- f g-+0 4"

0

1

1

x~-(1

1

-- x)" dx = f (--In x)" dx. 0

(4)

(In EULER [2, p. 52], he gives 1 • 2 . 3 ... p --- f e -v v p dr, the integral being taken from v = 0 to v = cx~.) He proceeds to verify (4) in some particular cases. Later EULER [3] generalized the factorial sequence in the form d : n = a(a H b) (a + 2b) ... (a + (n -- 1) b)

(5)

and this is what he (and his immediate successors) understood as a hypergeometric sequence. Corresponding to (1), he finds that for n, not restricted to integer values, "a + n'---~

" a + (n + 1) b \ ~ ]

aH2b (aH3b~" a _~S(n ~-"2) b \a + 2b] "'" '

(6)

and, in particular,

1 x2a4_b_1 ~ (d : 1)2 = a " [ " . , _ ~ dx. j o g V l - x .o

x2a_ 1

l/1

dx. -

-

X2o

(7)

22

J. DUTKA

In a later memoir, EULER found asymptotic values for expressions of the form (5),

etc. In connection with a memoir on deviations of planetary orbits published in 1749, EULER introduced the consideration of the expansion of functions of the form (1 -- 2a cos 0 + a 2)-s in terms of series involving the cosines of multiples o f 0 with 0 < a < 1. Thus " ba(-J) ---- b~j) b~(J) cosjO,

(1 -- 2a cos 0 + a2) -~ = ½ ~

(8)

j~--O0

where the coefficients

1 12 cosj0 dO ½ b(J) = ~ ; (1 + a 2 - 2a cos 0)*

(9)

termed "the LAPLACEcoefficients" in astronomical perturbation theory, have been the subject of intensive analytical and numerical investigation for more than two centuries. About 1766, J. L. LAGRANGE obtained a series development of b~J)/2 in a power series in a by writing (1 + a 2 -- 2a cos 0) -s = (1 -- aei°) -~" (1 -- ae-i°) -s

= ( ~ ( s + p - - 1 ) -a P 1e i P ° ) ' ( ~ (

s -- 1

aqa-iq°)

whence

(s+p--1) (s+P+J--1) s--1

h(J)

(s+j--1)aj

(s+p--1)(s+P+J--1)a2P/( p p+j s--1

EULER [4] considered in 1778 a series expansion of the form a. b

/ / ( a + 1) (b + 1) x 2

s=l+~x+

2(c+1)

/ / ( a + 2) (b + 2) x 3

-~

3(e+2)

+...

(I 1)

where H in each term denotes the coefficient of the power of x in the preceding term. This is, of course, the hypergeometric series (1.1). He shows that this series satisfies a differential equation equivalent to

d2s

x(1--x)~+[c

' (a+b+

ds

1) X ] ~ x - - a b s = 0 ,

(12)

History of the Hypergeometric Function

23

and that if one writes s = (1 -- x)"" z, then z satisfies a differential equation of the form d2z dz x(1--x)-~Tx2 + [c + (a + b -- 2e --1) x] ~x -- (e -- a) (e -- b) = O (13) on setting n = - - a - - b + e . Or, writing c - - a = ~ , e--b=fl, one obtains a differential equation satisfied by z by replacing s, a, b by z, o~, /5 respectively. In modern terms, s = (1 -- x)" z can be rewritten as F(a, b; c; x ) = ( 1 - - x )

....

bF(c--a,

c--b;

c; x),

(14)

one of the fundamental transformations of the hypergeometric function. ( C f the particular cases found by STIRLING, namely (3.1) and (3.2).) The foregoing memoir was the third of a series of four written by EULER in a four-week period in August-September 1778 concerned essentially with the evaluation of (9). By an incomplete induction on (=/2) Uh(p) n + l ~ he had previously found that cos pO dO (1 + a 2 2a c o s

/

-

0) n+l

-

Since

-

~a p

Pro) a2m"

('

(1 -t- a z -

(15)

m

2acos 0)"+1 --(1 -- a2)2~+1

.F(--n, p -- n; p + 1; a 2)

0

(16) which follows on substituting in (10) and (14). EULER also found on substituting --n -- 1 for n in (15) and applying (16) that f (1 + a 2 - 2a cos

0)" • cospOdO = ~ ( - - a ) p ~_a (n + m + p ) ( n - ~ - m ] a 2 m=o

0

m

m

\p + m]

p! a2,,' = ~(--a) p ( n ) 3 (n + m + p ) l (n + m)! \ P ~Z~-o (n + p) l nI (p + m) l = ~(-a)p

(

n

\P

F(n+p+l,

n+

1; p + l ;

a 2)

(17)

]

(cf (10)). Thus his result f cos pO dO o (1 + a 2 - 2 a c o s O ) "+1 (--1) p (n + p)n !! (n n ! -- p) l f (1 + a z -- 2a cos 0)" cos pO dO -- (1 -- a2)2n+l 0

follows.

(18)

24

J;, DUTKA

An integration problem relevant to the'development of the hypergeometric function was treated in EULER [5]. By writing (A + x~)z = [(A q- a')

x~)] ~ = (A q- an)a 1

(a"

A+

d

he expands the binomial fact0r in of x"-1(A q- x") dx and integrates term by term to obtain the equivalent of

" x") ~"dx a'' (A- @ a" f x"-'(A + =

0

n

=

(-1);

( 2p) [

an \p (

7'

p +

1) (19)

wl~ich reduces t o

o

of x ' - l ( A

:'

•

(

)

q- x " ) ~ d x = a m(A q-ma').F 1, --2; m+n 1; A q-a'a'/ ,

(19')

i

already determined by STmLINGin (3.3). In particular for ;~ = tz/n with /~ ~ m, n>0 and A = 0 , he finds from (19) m = 1 q '/~ + (/~ + n) m --'/J m q- n (m q- n) (m q- 2n)

#(/~ q- n) (# q- 2n) +... ' (m q- n) (m q- 2n) (m + 3n)

(20)

which appeared particularly remarkable since n appears on the right but not on the left. EULERverifies ths directly by subtracting the terms on the right, beginning with the first, from the left side successively and showing that the remainder thus obtained tends to zero. (The equivalent of 1

x--a

1

=--q

x

a

- -

x(x + l)

+

a(a q- 1)

x(x + l)(x + 2)

+ ....

for x > a > 0

had been given decades earlier by NICOLE, MONTMORT and STmLING. EULER'S result (20) follows from this.) Characteristically, EULER proceeds to investigate the conditions under which the verification method applied to the series. o~

o¢'B

A q- B--a q- C ~ ' b

a'fl'7

+ D ~ a . b . c q- "'"

(21)

leads to a valid result. If the sum of this series is denoted by s/t and the constants are defined by the relations

s = oc q- At,

a = fl q- Bt,

b = 7 + Ct, ...,

History of the Hypergeometric Function

25

then by successive subtraction from s/t, the values o~/t, o~. f l / t . a , t " a . b , ... are obtained and (21) converges if the infinite product

. " ~ "7/

..°

t

a

b

c

converges to a finite value H. (The value of the series is s/t -- AH, and H = 0 whenever A/o~ + B/~3 + C/7 + ... diverges.)

f xm-l(A + x") dx

EULER'S work on the evaluation of

was continued by

0

N. Fuss [1]. The method employed by Fuss is to set the indefinite integral equal to v • xm(A + x") ~ and to obtain a differential equation for v in terms of the variable z = x"/(A + xn). He then assumes that v has a power series expansion in z, whose coefficients are determined on substituting in the differential equation, and obtains EULER'S series (19'). In particular, for m = A = 1, n = 2 and 2 = --1, he gets the series x [ (x+___~x2) 2 " 4 [ x2 "~2 arc tan x -- -------5 1 + x 1 + ~ ]+ ~ \1----~5 ] 2"4"6 [

x2

-1- 5------~ 3• ~ 1

13

] + . . . J.

He also shows how the series s =--

+--. a

a

b

+

a

b

c

+ ...

(21)

...

(22)

can be transformed into a continued fraction

o~ s

--

=

a

a " fl b cO b+fl-c+v-d+•-

and applies this result to (20). Many writers on the hypergeometric function attribute to EULER a representation of the function in the form of a constant times a definite integral. As a justification for this, they point to [EULER, 6, Chs. 8-11] in which he investigates a particular class of second-order linear differential equations of the form

d2y dy x2(a + bx") ~x z + x(c + exn) -~x + ( f + gXn) y = 0

(23)

which includes, although unremarked by EULER, the differential equation of the hypergeometric function, (12), as a special case. In Chapter 9, he is concerned, inter alia, with transformations of (23) which yield other differential equations of the same form. (Cf. (12) and (13).) In Chapter X, Problem 130, he considers the solution of some equations of the form (23) by means of definite integrals

y = f V(u, x) dx in which V(u, x) is assumed to have special forms. In particular, 0

26

J. DUTKA

in ¶ 1033, EULER shows that C

Y = f x~-l(u 2 -k x2)~ (c 2 -- x2f dx

(24)

0

is a solution of d2y u(c~ + u2) d-~ -- ((n + 2~, - 1). (e 2 + u2) + 2(~ +

~) ~)

dy

+ 2/z(n -/2/z + 2~,) uy = 0

(25)

which is not of the form of the hypergeometric differential equation (12). (It can readily be shown, however, that if w ---- e2/(u 2 + e2), then y=const.w

(--/~,

-~.F

.

~+1,

-~--t-~,-I-1;

w)

is a solution of (23).) The earliest integral representation of the hypergeometric function was given in A. M. LE~ENORE'S [1, ¶ 114] investigation of some integrals. He found 1 f x ' - - l ( 1 --x)--r(1 + a x ) - " d x = B ( p ,

1 --r)

o

(n,

×F

l--r,

p--r+l"

'a+l

a)

(26)

but he did not relate this evaluation to EULER'S series (11). In a little known doctoral dissertation of 1833, P. C. O. VORSSELMANDE HEER [1, 10-12] showed in the notation of GAUSS' memoir of 1813 that 1

f z~-1(1 F(~, /3, ;~, x ) = o

-- z) e-a-1 (1 -- xz) -~ dz

1

f z~-I(1

(27) -- z) e-~-I dz

0

(for Ixl H 1) by expanding the binomial (1 -- xz) -~ in a series and integrating term-by-term to obtain a power series in x whose coefficients involve ratios of beta functions. (He also developed many of the results previously obtained by others for the hypergeometric function from this representation.) Three years later, E. E. KUMMER [1, ¶ 27] obtained (27) similarly.

5. The work of Pfaff

JOHANN FRIEDRICHPFAFF (1765-1825) was one of the leading mathematicians in Germany in the latter part of the eighteenth and early nineteenth centuries, and is perhaps best known for his work on differential equations. Beginning in the late 1790's, he was the teacher and friend of C. F. GAUSS at Helmstadt, but it is not known how much influence he exerted on the latter's mathematical work.

History of the Hypergeometric Function

27

In 1797, he wrote a monograph in which a long memoir (PFAFF [1]) was devoted to researches on the integration of the inhomogeneous equation

d 2y -~x x2(a -k bx n) ~x 2 -1- x(c -t- ex n) @ ( f -k gx") y = X

(1)

where X is a function of x. (Cf. (4.23).) The methods which he follows are essentially a further development of those given by EULER [6, Chs. 7, 8]. The power series solutions which are obtained by solving second order differential equations are called by him "hypergeometric." (A proposed second volume in which these concepts were to be further developed was not published.) PFAFF supplements nine special cases considered by EUL~R by initially transforming the homogeneous form of (1) with n = 1 into equations of the same form with f = 0 in three ways by means of the substitution y ---- xP(a -t- bxn) q" v. (The resulting equation is essentially equivalent to (4.12).) On pp. 155-157, by a proof which involves two series representations of the same function with complicated coefficients, he shows that (

e

(a + bx")rF --r,

c

c

n b + na -t- 1 -- r; na q-

(

e2p--l_ = arF --r, -~-1n

2p--1

__c2p--l_ -}- r; na + n

-k 1; a

n

b~x,/)

(2)

bXa") -f-1;

--

where r is a positive integer. This is a particular case of the relation

F(a, b; c; x ) = ( 1 - - x ) - a F

(

a, c - - b ;

e;

(2')

x--1

for [x/(x( -- 1) I < 1. The relation follows from (4.27) on writing (1 -- xz) = (1 x) [1 -- x(1 -- z)/(x -- 1)] in the numerator on the right. A particular case of this had been obtained earlier by STmLING in (3.1) and (3.2). Corresponding to (4.9), LEa~NDRE [1, 55 partie, 278] showed that -

-

½b~J)=( s+j-J.

__--

(

s@j--1, j

1) aJ F(s, s + j; j + l', 02)

)

aJ(1--a2)-~F

(

s, l - - s ;

j-I-l',

a2

--1

(3)

and the equivalent of (2') was also proved independently by C. GtJD~RMANN in 1830. On p. 160, PFAFF shows, by applying series methods, that if r' is an integer

F(--r, k - - r ,

t,

k + l ; --fix") = (1 ~- 13x")l+r+r'F(r -t- k + 1, r" q- 1; k + 1; --fix ~)

(4)

which is a particular case of the relation

F(a, b; c; x ) = ( 1 - - x ) c - a - b F ( c - - a , previously developed by EULER (4.14).

c--b;

c; x)

(4')

28

J. DUTKA

This relation follows from (2') as shown by VORSSELMANDE H~ER [1], for if v = x/(x -- 1), F(a, c -- b; c; v) = F(c -- b, a; c; v) = (1

-

v)b-e F ( c - - b, e -- a;

-

c; x)

and F(a, b; c; x) = ( 1 - - x ) - a . ( 1 - - x)C-b F(c -- b, c - - a ;

c; x).

An important, but relatively little known, memoir was published in 1797 (PFAFF [2]). (A summary of some of the results of this memoir was given by R. ASKEr [1].) PFAFV initially considers a problem previously treated by EULER [5] concerning the evaluation of f xm-l(A + x")~dx and develops a number of series solutions including EULER'S. Then making use of a theorem on series expansions of EULER'S, he obtains another version of (4) F(--r,

q; p;

- - ~ x n) -~- (l -~- f l x n ) r F

(

p - - q; p;

--r,

(5)

1 + t3xn ]

from which he obtains several corollaries, and for r a positive integer F(--r, q; p; --x) =p(p÷r)--F(-~_~F(--r,.. P(p) F(p + r -- q) ..

q; q - - r - - p + l ;

1 +x).

(6) (This would now be obtained from a standard textbook result for linear transformations of hypergeometric functions P(c) I'(e -- a -- b)F(a, F(a, b; c; z ) = p ( c _ a ) i , ( c -~ b; a + b - - c + l ;

+(1--z)C-=-bI'(c)F(a÷b--C)F(c--a, F(a) r(b)

c--b;

c--a--b+

l--z)+ l;

l--z)

where here the second factor on the right vanishes since lira 1/_P(a)-+ 0 as ¢ / ~ --r.) On setting x = 1 in (6), PFAFF obtained the result F(--r, q; p;

F(p) ~(p + r - - q ) 1)= f,(p+r) F(p_q)

(7)

which is usually called VANDERMONDE'Stheorem, after the eighteenth-century French mathematician, but which was k n o w n before him. PFAFF, on pp. 46-48, introduces what were later called contiguous functions by GAUSS. After defining f(q, p, r) and F(q, p, r) by

f(q, p, r ) = F ( - - r ,

q; p;

--x),

I'(p) I~(p + r - - q ) F(q, p, r ) - - I ' ~ p - - q ) I~(P+ r) F(--r, q; 1 ÷ q - - p - - r ;

1 +x)

(8)

History of the Hypergeometric Function

29

respectively where from (6), f(q, p, r) = F(q, p, r) for r a positive integer, he shows that f(q, p, r) = f ( q ,

p, r -- 1) + q x f ( q

+ 1, p + 1, r -- 1).

(9)

This follows on eliminating the left hand member of the equations d

~Z

[(z)--r F(--r, q; p; z)] d = - - r ( z ) - r F ( 1 - - r , q; p; z)--~z[(Z)-rF(--r, q; p; z)]

rqz -r = - - r ( z ) - ~ - l F ( - - r , q; p; z ) - - ~ F ( 1 P where z ---- --x. The equivalent of t.

.

F ( - - r , p + r , p, --13x") (1 + fix") r

F(

_r I

- - r , 1 + q; 1 + p ;

z)

, p + r;p; --fix")

(1 + ~x")"

was also proved by PFAFF on pp. 48-49. It follows immediately from the observation that from (2) each of the terms in (10) equals F(--r,

--r'; p; ~x"/(1 + ~x")).

Perhaps the most remarkable result in PFAFF'S memoir is that given on pp. 51-52 to the effect that if l is a positive integer

,fro p,

]

--(l+r+m+p--1);1 P(p + r + l). ]~(p)/'(p + m + 1) (p -k r + m) = F(p+l) P(p+r) ]'(p+m)(p+r+m+l)

(11)

which lie proves by induction on l. This is, of course, usually known as SAALSCH~3TZ' theorem after the German mathematician who rediscovered it more than ninety years later, although the relevant work of EULER and PFAFF was mentioned by JACOm and in a well known textbook by E. HEINE [1, Bd. 2, 357-398] on spherical harmonics. 6. The work of Gauss

The evolution of the term "hypergeometric" from a description of a particular type of sequence by WALLIS to the power series solution of a particular kind of second order linear differential equation by PFAFF finally reached modern form in the work of GAUSS, a student and friend of PFAFF'S. GAUSS' contributions are in the form of one published memoir (GAuss [1]), a second memoir in essentially finished form, which only appeared posthumously (Gauss [2]), and additional material in correspondence, notebooks, etc., beginning in 1805 (GAUSS [3]); much of this came to light decades after GAtlSS' death in 1855. The material has

30

J. DUTKA

been summarized, in a masterly fashion, by L. SCHLESINGER[1]. It was later supplemented by a report by A. I. MARKUSCHEWITSCH [1] in a memorial volume on GAUSS. In GAUSS [1], relatively few references to the work of GAUSS' predecessors are given, so that it is often difficult to distinguish between what GAUSS learned from earlier investigators and what he himself discovered or rediscovered. The introduction contains some remarks concerning notation, in which GAUSS defines a series c~fl F(o¢, fl, 7, x ) = l - k l , T X - 1 -

~x(~x-4- 1)/3(/3 4- 1) 1 " 2 ( 7 - k 1) xxq-etc.

(1)

in which EULER'S notation (4.11) is essentially followed. GAUSS then considers the convergence of the series for complex values of x and finds that the infinite series converges for Ixl < 1 and diverges for ix] > 1. Some twenty-three developments of algebraic, logarithmic, hyperbolic and trigonometric functions expressible in terms of the series (1) are then presented. Paragraph 6 is devoted to the series (aa -k bb -- 2ab cos ~)-" = / 2 = A + 2A' cos ~"-k 2A" cos 2~ + 2A'" cos 3~" + . . .

(2)

in which results previously obtained by J. L. LA~RANGE are rederived. (Cf. (4.10).) In addition, GAuss obtains the new results A Cp) =

P

(a 4- b)-(z"+2P) (ab)P. F

q-- p, p q- ½, n + ½, 4- (a -~ b)21

(3)

which stem from expanding (2) in the form f2 • [(a -b b) 2 4- (1 4- cos 0 ] - "

and evaluating f cos pC.

(cf. (4.9).)

0

The first section is concerned with relations among contigouus functions. By a contiguous function to F(o~, b, 7, x), he means a function in which one of the first three parameters is increased or diminished by unity, so that there are six contiguous functions associated with F. He shows that a linear relation holds between any two of these functions and F, and there are fifteen such relations. The proof consists in determining the equality which exists between coefficients of x m (for m a general non-negative integer) in the three terms. The process may be continued to obtain linear relations between F and the functions F(o~ + 2,/3 -k/z, ), -k v, x), F(~ + 2',/3 q- #', 7 -k v', x) in which 2,/l, v, 2', #', v' assume the values 0, + l , or --1 and the number of relations increases to 325. (GAUSS' Equation [18] includes (4.9) for particular values of the arguments.) The second section is concerned with the continued fraction developments of F(o¢,/3 -k 1, 7 + 1, x)/F(o~,/3, 7, x) and the important special case F(o¢, 1, 7, x). The method employed stems from that of J. H. LA~mERT [1] who obtained an algorithm applicable to the quotient of power series analogous to the Euclidean algorithm for obtaining the largest common divisor of two integers. (LAMBERT

History of the Hypergeometric Function

31

applied the method to tan x = sin x/cos x and to tanh x/2 = (e~ -- 1)/e~ -k 1) and proved the convergence of the respective continued fractions obtained by the formal algorithm.) GAUSS' procedure, which does not include a consideration of the convergence of the continued fractions, is similar to a more refined Version of LAMBERT'Smethod developed by A. M. LEGENDRE,which appeared as a note in the numerous editions of the latter's textbook on geometry. The question of the convergence of GAUSS' continued fractions was later investigated by B. RmMA~N and E. HEINE, but not finally settled until the work of L. W. THOMAE [1] in 1867. The results obtained by GAUSS include a large class of cases which arise frequently in applications and are often particularly useful in obtaining numerical approximations for the values of the functions. Examples of continued fraction expansions are given for (1 + u)", log (1 + t), log (1 -k t)/(1 -- t), et etc. The third section begins with a remarkable discussion of the convergence of the series F(~, fl, 7, 1) which shows that GAUSS' approach to the underlying problems was far in advance of his contemporaries. A detailed discussion will not be given here since the convergence of the hypergeometric series is treated in standard texts as well as in more specialized works such as REIFF [1, 161-167]. The remainder of the memoir is primarily concerned with the gamma function and with the derivative of the logarithm of this function. GAUSS defines a function 1.2.3...k H(k, z) -- (z + 1) (z -F 2)... (z + k)

• k z

(4)

and H(z) = lim II(k, z)

(4')

k-->- oa

which is a generalization of the factorial function (cf EULER'S definition (4.3)) and shows that the functional equation /-/(z q- 1)-----(z -F 1)//(z)

(5)

holds. (GAuss' notation has not survived. H(z) is usually replaced by LUaENDRE'S H(z + 1) or sometimes by z! in the modern literature.) Sometime after this memoir was printed, GAUSS appeared to have changed his mind regarding his approach to the factorial function and stated "the best definition of H(m) is that +oo

II(m) --- f

e(m+l)x e-e x d x " ,

(6)

--oo o~

obtainable by setting t = ex in the integral F(rn + 1) = f tree-t dt. But this o definition appears to have been used only in an article by J. LIOUVILLEin 1852. GAUSSinvestigates the behavior of H(z) for real z and furnishes a table of numerical values of log II(z) and W(z) = dlog H(z)/dz at the end of the memoir. Having previously established from one of his linear relations for contiguous functions (Eq. [15]) that F(~x, /3, ~,, 1 ) - -

(~ - " ) ~,(~, - .

(r

-

fl) F(~x, fl, 7 + 1 , fl)

1),

32

J. DUTKA

GAUSS replaces 7 by y + 1, 7 + 2, ... successively and shows that

H(k, 7 - - 1 ) ' H ( k , 7 - - ° ~ - - / 3 - - 1 ) F(a, /3, 7, 1 ) = //(k, 7 - - o ~ - - l ) ' / / ( k , 7 - - / 3 - - 1 ) F ( a , He then lets k - + ~

/3, r + k, 1).

and obtains the important result

//(7 -- 1)/7(7 -- o~ --/3 -- 1) F(o~, /3, 7, 1) = ii(7 _ o¢ -- 1)/I(7 - - f l -- 1)

(7)

(where R0' -- o~ --/3) > 0). (Cf (5.7).) GAUSS proceeds to apply his definitions and results to derive known properties of the gamma function previously obtained by EULER, as well as the remarkable product formula

nnZH(z).//(z - 1 )

.//(z -2)

... H ( z

n--n i.)

u(nz)

(2z0-T-n-'

¢-y

(8)

(Eq. [57], the special case n = 2 having been obtained previously by LEGENDRE). GAUSS then shows that for 2, # > 0,

o

xZ-l(1 -- x~)" dx =-'-ff F --v, ~ ,

)

~ + 1, x t') /~

/z

(9)

and that for the upper limit x = 1

II~.~...Hv #

1

of x ~ - ' ( 1

- x")" d x -

(9')

2r/(~-Z + ~)

and compares this with the results which EULER had laboriously obtained. (Cf however (3.2).) The memoir to which GAUSS refers is presumably EULER [7], which was published in 1789. On pp. 419-420, EULER expresses a beta function as an infinite series. I.e.,

i 0

xP-'dx

--1F(

I/(1 -- xn)n--q

p

1 -q'

p"

n

p+I;

1)

n" n

etc. GAUSS confirms certain numerical results associated with the rectification of the lemniscate and expressible in terms of / / ( ± 1 / 4 ) and /7(--1/2) with results previously obtained by STIRLING [1]. To deduce an integral representation for the factorial function, GAUSS (p. 15i) notes that

H2 . H~, _ f y~_~ l

z~(2 +,,)

o

Yydy

T ] -~

and since as v-+ (x~, (1 --y/v)~-+ e -y, h e finds from [he definition of II(v, 2)

History of the Hypergeometric Function

33

that

lim

~,~H,~ " H~

~-+o~ ~H(,~ + ~)

II(v, ~) -- lim - - ~-~o~

2

f y~'-le-y dy.

(10)

o

(He does n o t justify the limit process on the right.) After some remarks on an extension of the STIRLING expansion, which he attributes to EULER, to log H(z), GAUSS remarks on the divergence o f this series and mentions that the BERNOULLI numbers (which appear as coefficients in the series) constitute a "hypergeometric series." (Evidently GAUSS is still using this term in its pre-PFAFFian sense !) The last part o f the memoir is devoted to the development o f important and fundamental results associated with properties o f ~ ( z ) = d log H(z)/dz. ( N o w adays 7t(z) is usually defined as the derivative o f In I'(z).)But since this function is not directly associated with the hypergeometric function, these results will not be discussed here.

Acknowledgment. I express my appreciation to Professor RICHARD ASKEY, University of Wisconsin, and to Dr. W. KAUFMANN-BI)HLER,Springer-Veflag, New York, for their interest in the manuscript and for helpful comments.

References ASKEY, R. [1], "A note on the history of series," MRC Technical Summary Report @ 1532, March 1975, University of Wisconsin. BINET, JACQUESP. M. [1], "M6moire sur les int6grales d6finies eul6riennes," Journal de l'Ecole Polytechnique, Cahier XXVII (1839), 123-343. DAVIS, PHILIP J. [1], "Leonhard Euler's Integral .... " American Mathematical Monthly Vol. 66 (1959), 849-869. EULER, LEONHARD [1], "De progressionibus transcendentihus ...," Opera Omnia, Ser. 1, Vol. 14, 1-24, Leipzig and Berlin, 1925. EULER, LEONHARD [2], "De curva hypergeometrica hac aequatione expressa y = 12 • 3 . . . . x," Opera Omnia, Ser. 1, Vol. 28, 41-98. EULER, LEONHARD [3], "De termino generali serierum hypergeometricarum," Opera Omnia, Ser. 1, Vol. 16, Part 1, 139-162. EULER, LEONHARD[4], "Specimen transformationis singularis serierum," Opera Omnia, Ser. 1, Vol. 16, Part 2, 41-55. EULER, LEONHARD [5], "De resolutione formula integralis f x m-a dx (A + i n ) z . . . . " Opera omnia, Ser. 1, Vol. 19, 110-128. EULER, LEONHARD[6], Institutionum Calculi Integralis Vol. II, Petropolis, 1769, reprinted in Opera Omnia, Ser. 1, Vol. 12. x p-1 dx r EULER, LEONHARD [7], "Comparatio valorum formulae integralis J xn)n--q

¢(1

a termino x = 0 usque ad x = 1 extensae," Opera Omnia, Ser. 1, Vol. 18, 392-435. Fuss, NICOLAS [1], "De resolutione formulae integralis f x m-a dx (A + in) ~ ...," Nova Aeta Aead. Seient. Petropolitanae, T. XV (1799-1802), (1806), 55-70. GAUSS, C. F. [1], "Disquisitiones Generales Circa Seriem Infinitam ...," (1813), Werke~ Bd. 3, 124-162. GAUSS, C. F. [2], "Determinatio seriei nostrae per aequationem differentialem," Werke, Bd. 3 (1876), 207-229.

34

J. DUTKA

GAuss, C. F. [3], "Zur Theorie der unendlichen Reihe F(c~, fl, 7,, x)", Werke, Bd. 10, Teil 1, (1917), 326-359. HEINE, E. [1], Handbuch der Kugelfunctionen, 1878, reprinted WiJrzburg, 1961. KUMMER, E. E. [1], "Lrber die hypergeometrische Reihe .... " Journal fiir die reine and angewandte Mathematik, Bd. 15 (1836), 39-83, 127-172. LAMBERT, J. H. [1], "M6moire sur quelques propri6t6s remarquables des quantit6s transcendantes circulaires et logarithmiques," MOmoires de l'AeadOmie des sciences de Berlin, [17] (1761), 1768, 265-322. LEGVNDRE, A. M. [1], Exercises de Calcul lntdgral, II, Quatri6me Partie, Sect. II, Paris, 1816. MARKUSCHEW~SCH, A. I. [1], "Die Arbeiten von C. F. Gauss fiber Funktionentheorie" in C. F. Gauss: Leben and Werk, Gauss Gedenkband, ed. by H. REICHARDT, Berlin, 1955, 151-182. NEWTON, ISAAC [1], The Correspondence o f .... Vol. II (1664-1660, Cambridge, 1960, ed. by H. W. TURNBULL. N~WTON, ISAAC [2], The Mathematical Papers o f . . . , Vol. VII (1691-1695), Cambridge, 1976, ed. by D. T. WmTESIDE. PFArr, J. F. [1], "Nova disquisitio de integratione aequationis ...," Disquisitiones Analytica, I, 135-224, Helmstedt, 1797. PFAFr, J. F. [2], "Observationes analyticae ad L. Euleri Institutiones Calculi Integralis, Vol. IV, Supplem. II et IV, Nova Acta Academiae Scientiarum Petropolitanae, Tome XI (1797), Histoire, 38-57. REIFr, R. [1], Geschiehte der unendlichen Reihen, Tfibingen, 1889. SCHLESINGER, L. [1], "Ueber Gauss' Arbeiten zur Funktionentheorie," Carl Friedrich Gaass Werke, Bd. 10, Teil 2, (1922-1933), Nr. 2. SCRmA, CHRISTOPrt J. [1], "Von Pascals Dreieck zu Eulers G a m m a Funktion .... " Mathematical Perspectives, New York, 1981, ed. by J. W. DAUBEN, 221--235. STIRLING, JAMES [1], Methodus Differential& .... London, 1730. TrIOMA~, L . W . [1], "Ueber die Kettenbruchentwicklung der Gauss'schen Quotienten F(c~, fl + 1, ~ + 1, x)/F(~, fl, 7, x)," Journal fiir die reine und angewandte Mathematik, Bd. 67 (1867), 299-309. TWEEDIE, CHARLES [1], James Stifling .... Oxford, 1922.

VORSSELMAN Dr HEER, P. E. O. [1], "Specimen inaugurale de fractionibus continuis," Utrecht, 1833. WALHS, JOI~N [I], Arithmetica tnfiuitorum, 1656, Opera Mathematica, Vol. I, Oxford, 1695. WALHS, JOHN [2], A Treatise o f Algebra .... London, 1685. Audits & Surveys, Inc. One Park Avenue New York (Received February 3, 1984)