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3 MPLE J ROOF OF AMANUJAN S 1 YMMATON OF THE / n rewj LTT; H Mathematics Research ent Universit y of Wisconsin Ma dison 810 Walnut Street Madison, Wisconsin AG i NSF r i pptovsd fs i pubhc rebu s Distr ib utis us uuli uded Sponsored by U. S. Army Research Office end National Science Foundation P. O. Box Washington, D. C Research Triangle Park North Caroline 27709

4 UNVERSTY OF WSCONSN MADSON MATHEMATCS RESEARCH CENTER A SMPLE PROOF OF RAMANUJ AN S SUMMATON OF THE George E. Andrews and Richard Askey Technical Summary Report 1669 August ABSTRACT A simple proof by functional equations is given for Ramanujan s sum. Ramanuj an s sum is a useful extension of Jacobi s triple product formula, and has recently become mportant in the treatment of certain orthogonal polynomials defined by basic hypergeometric series. AMS( MOS) Subj ect Classification: 33A25, 33A30 Key Words: Basic hypergeometric functions, Theta function s, Ramanujan sum, Jacobi triple product. Work Unit No. 2 (Other Mathematical Methods) \ \ /. * \ \... L Sponsored by the United States Army under Contract No. DAAG2975C24, and by the National Scienct Foundation under Grants and MPS A02. it T T 1j

5 A SMPLE PROO F OF RAMANUJAN S SUMMATON OF THE George E. Andrews 1 arid Richard Askey n S; p. 222, eq. ( )] G. H. Hardy alludes to Ramanuj a ) S remarkable formula with many parameters. (a;q) () v n ( a q,x ) (b;q) 11 b n = (b/a,q) (q;q) (q/ax;q) (ax;q) (b;q), (b/ax;q) (q/a ;q) (x;q) where b n < xl <1, q <1, (a;q) (1aq ), and (a;q) = (a;q) /(aq ;q) ). fl There are four published proofs of this result ([1], [2], [4J and [7]). Those in [1 J, [2] and [7 J rely on somewhat tricky rearrangement of series and on the q anaiog of Gauss s summaton [10; p. 97, eq. ( )] (2) (a;q) (b;q) ( ) (c/a ;q),, (c/b;q) n=0 (c;q )( q ;a) (c;q) (c/ab;q) where c < min (l, lab ). The other proof uses the qanalogue of the binomial series [10; p. 92, eq. ( )]: (3) f ( (T i = 1. n=0 l t l < i, qj< i, but it is far from simple. Since Ramanujan s summation (1) has rec ntly become important in the treatment of certain orthogonal pol ynom ials defined j Sponsored by the Uni ted States Arm y under Contract No. DAAGZ97 5C24, and by the National Science Foundation under Grants and MPS A02.

6 ry basic hypergeometric series (3J. it has become wort hwhile to present r. ilmost trivial proof of 1). Another very simple proof has n fou nd by M. smail [6J. Proof of (A) We begin by noting that for <1. 1(b) is an analytic function of b nside Jb mm (1. ax ). since b n ( b n )... ( ) x (a;q) x i q A Ci t. fl q ( t) i j) lb.nl n=0 n n (1 a a )... ( q q Furthermore, a;q, x a;q,qx (), b a 1 b 1 ( b (a;q) 31 x (b; q) = x 1 (l b q (a;q ) 1 x 1 ( Hence (6) f( bq ) x 1 ( lb)f( b) a l b x ( q ) 1 tj 1 ( b/q ii (a,, ) q x (a:q) (l bq = a b = eb (lb) f(b) + eb m f( bq ) and so or (1 (7) f(b) = )f ( bq) (1 b )(x ab 4 ) f(b) b (1 ( b)(l f we iterate (7) n times we find that b 2 y..

7 ( 8) f(b) (b/ a ;q) (b;q) (b/ax;q) f lxñ and since f(b) s analytic n the neighborhood of 0 given by bj < ax. we obtain in the limit as n. (b/a ; q) f(0) (9) f(b) = (b;q) (b/ax;q) Now we observe from (4) and (3) that (10) f(q) (a;q) x (ax, q) n=0 (q;q) = (x;q ) This allows us to evaluate f(0) by setting b q in (9) ( ) f(0) = (q;q) (; q) f(g) (q/a; q) (q;q) ( ;q) ( ax;q) (q/a;q) (x;q) Finally we may utilize (11) to eliminate f(0) from (9) 12 as desired. a; q,x f b b (b/a ;q) (q;q) (q/ax;q) (ax;q) (b;q) (b/ax;q) ( q/a,q) (x;q), Note that Jacobi s triple product identity follows directly from (1) if we replace a by a, x by za and then set a = b = 0: ( 13) n n(nl)/2 n n oo () q z = (q;q) (q/z;q) (z;q) 3... riislc _

8 (a;q). J. Schoenberg has pointed out an interesting property of (b;q) which follows from Ramanuj an s sum. A sequence n = 0, ± 1,... is said to be totally positive if all subdeterminants of the doublely infinite matrix A = <, are nonnegative. Schoenberg [9] proved that a sequence a function f(z) S. fl is totally positive if the bilateral generating az has the representation l (1 + a. z)(l + 6z ) cz+d z (14) f( z) = e i i 1=1 (1 3 z)(l,r c, d, a 1, 1, 0, (a ) < in the nterior of an annulus centered at the origin. and so f a < b < 0 in (1) then, the generating function has the form (14) = (a; ) [ (1 k bg )(l pg ) (b;q) = i k=o (1 aq )(l k ) is a totally positive sequence f or a < b< 0, 0 <q <1. Schoenberg [9] proved this when b = 0. For an extended discussion of totally positive sequences see Karlin [8].

9 References i. G. E. AnJrews, On Ramanujan s summation of American Math. Soc., 22(1 969), Z i ( a; b; z), Proc. 1. G. E. Anjrews, On a transformation of bilateral series with applicati ns, Proc. American Math. Soc., 25 (1970), C. E A drews a n. R. Askey, Monograph, to appear. 4. W. Hahn, Be:tr ge zur Theorie der Heinescheri Reihen, Math. Nach., 2 (19 4 ), C. H. Hardy, Ramanuj an, Cambridge University Press, Cambridge, 1940 ( Reprinted : Chelsea, New York). (.. M. smail, personal communication. 7. M. Jackson, On Lerch s transcendant and the basic bilateral hypergeometric series J. London Math. Soc., 25 (1950), S. Karlin, Total Positivity, Volume One, Stanford University Press, Stanford, J. Schoenberg, Some analytical aspects of the problem of smoothing, Studies and Essays Presented to R. Courant on his 60th Brthday, nterscience Publishers, New York, 19 48, L. J. Slater, Generalized Hypergeornetric Functions, Cambridge University Press, Cambridge, Footnotes: p 1a11y sponsored by NSF Grant and by the United States Army under Contract No. DAAGZ975C24. (Z) partiaiiy supported by NSF Grant MPS A02 5 Jr

10 U NCASSFED T W, $ NAGE ewn, D.s a Sni.r.d) $Cu* t C L A $ 1, P CA T, O M 0? r msno.t.t Tt? N U M? U rj 6 9 TY P E OF NPo NT A p N,0o COVENED Summary Report no specific reporting period PENPORMNG ORG D. PNFORMNO O N GA N,ZA ON N A M E AND AD DRESS Mathematics Research Center, University of 610 Walnut Street Wisconsin Mad ison,wisconsin PROGRAM ELEMENT, PROJ ECT, AS,c A R A S WO RK UNT NUMARS 12. REPORT O A T S CONTROLLNG OPPCE NAM E AND ADDRESS August 1976 See tem 18 below t DMG2975 C24 MPS A George E. Andrews and Richard Askey RPONT NUMD. C O NT NA C T ON G RA NT WUMAER(. : A U T $ O R(.) S N C P S W T $ CA t ai.oo NuMSR S (W,d SubfSf ) A SMPLE PROOF OF RAMANU JAN S SUMMA TON READ tn STRVCTON$ B1?oR COMPL.ETUO FORM V CN 2. GOVT ACC!UON NO.. *5. MO,l,!O *,NZ(NCY NAME S AbOR1 $ è f%spiffl tvui Con&olSng Offic.) NUMSEN OP PAGS SECURTY C. ASS. (of ffif,.porf) U NC.ASSFED US.. DEC.*$$pCATONQQWN ONAD,N SCHEDULE OSTR5UTON STA t EMEN T (.1 mi. R.pcr l) fa. Approved for public release; distribution unlimited. *7 DSTRSUTON ST A T EM ENT (.f A..b.r.c S. $UN* LSMSNTARY NOTES Rp,t.,.d,. S. k 30. U du.q if S. ) U. S. Army Research Office and National Science Foundation P.O. Box Washington, D. C Research Triangle Park, N ort h Carolina L KEY WORDS (C.øCMu.n, v.r d d.nufy by block.w sb.v).sd? n.e y. Basic hypergeometric function s, Theta func ti on s, Ramanujan sum, Jacobi triple product 20., ev, Sd. U n. c.. w *nhf by bs..b, S.c) AU. A S T R A C Y 4. A simple proof by functional equations is given for Ramanuj an s 1 l sum. Ramanujan s sum is a useful extension of Jacobi s triple product formula, and has recentl y become important in the treatment of certain orthogonal polynomials defined by basic hypergeometric series. DO J A M 73 : , TON OP NOV AS S OSOLEt E UNCLASSFED S ECURTY CLAU P$CATON OP t ll S PAGE (1Ssn b.. Eft(..d) s