We give a new proof of the classical Watson theorem for the summation of a 3F2 hypergeometric series of unit argument. The proof relies on the two well-known Gauss summation theorems for the 2F1 function

Paris, R. B.

Rathie, Arjun K.

English

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Applied Mathematical Sciences, Vol. 3, 2009, no. 4, 161 - 164A New Proof of Watson’s Theoremfor the Series 3F2(1)Arjun K. RathieManda Institute of TechnologyRaisar NH-11, Jaipur Road, Bikaner 334001, IndiaR. B. ParisDivision of Complex SystemsUniversity of Abertay Dundee, Dundee DD1 1HG, UKR.Paris@abertay.ac.ukAbstractWe give a new proof of the classical Watson theorem for the summa-tion of a 3F2 hypergeometric series of unit argument. The proof relieson the two well-known Gauss summation theorems for the 2F1 function.Mathematics Subject Classification: 33C20Keywords: Watson’s theorem, hypergeometric series1. IntroductionThe classical Watson theorem for the summation of a 3F2 hypergeometricfunction of unit argument takes the form3F2(a, b, c12a + 12b + 12, 2c; 1)=Γ(12) Γ(c + 12) Γ(12a + 12b + 12) Γ(c + 12− 12a − 12b)Γ(12a + 12) Γ(12b + 12) Γ(c + 12− 12a) Γ(c + 12− 12b)(1.1)provided Re(2c − a − b) > −1 and the parameters are such that the serieson the left is defined. The proof of this result when one of the parameters aor b is a negative integer was given by Watson in [7], and subsequently wasestablished more generally in the non-terminating case by Whipple in [8].162 A. K. Rathie and R. B. ParisThe standard proof of the general case given in [2, p. 149; 6, p. 54] relieson the following transformation due to Thomae3F2(a, b, cd, e; 1)=Γ(d) Γ(e) Γ(s)Γ(a) Γ(b + s) Γ(c + s)3F2(d − a, e − a, sb + s, c + s; 1),where s = d + e − a − b − c is the parametric excess, combined with Dixon’stheorem for the evaluation of the sum on the right when d = 12a + 12b + 12and e = 2c. An alternative and more involved proof [4, p. 363] exploits thequadratic transformations for the Gauss hypergeometric function. A thirdproof, due to Bhatt in [3], exploits a known relation between the F2 and F4Appell functions combined with a comparison of the coefficients in their seriesexpansions.In this note, we give a simple proof of (1.1) that relies only on the well-known Gauss summation theorems for the 2F1 function, namely [1, pp. 556,557]2F1(a, b; c; 1) =Γ(c) Γ(c − a − b)Γ(c − a) Γ(c − b) , Re(c − a − b) > 0 (1.2)and2F1(a, b;12a + 12b + 12; 12) =Γ(12) Γ(12a + 12b + 12)Γ(12a + 12) Γ(12b + 12). (1.3)We shall also require the following elementary identities for the Pochhammersymbol, or ascending factorial, (a)n = Γ(a + n)/Γ(a) given by(a)2m = 22m(12a)m (12a + 12)m, (a)k+2m = (a)2m (a + 2m)k (1.4)for nonnegative integers m and k, together withLemma 1 Let k be a nonnegative integer and c be (in general) a complexparameter satisfying 2c = −1,−2, . . . . Then(c)k(2c)k=k/2∑m=02−k−2mk!(c + 12)mm! (k − 2m)! , (1.5)where k/2 is the integer part of k/2.The proof of this lemma uses (1.2) to express the ratio of Pochhammersymbols as a terminating Gauss hypergeometric function in the form(c)k(2c)k= 2−k2F1(−12k, 12 − 12k; c + 12 ; 1)= 2−kk/2∑m=0(−12k)m (12− 12k)m(c + 12)m m!.Watson’s theorem 163The result (1.5) then follows upon making use of the identity(−12k)m (12− 12k)m =2−2mk!(k − 2m)! .2. Proof of Watson’s theorem (1.1)We denote the left-hand side of (1.1) by F and express the 3F2 function as aseries to findF =∞∑k=0(a)k (b)k (c)k(12a + 12b + 12)k (2c)k k!=∞∑k=0(a)k (b)k(12a + 12b + 12)k k!k/2∑m=02−k−2mk!(c + 12)m m! (k − 2m)!by Lemma 1. Upon reversal of the order of summation, making use of theeasily established result [5, p. 57]∞∑k=0k/2∑m=0A(m, k) =∞∑m=0∞∑k=2mA(m, k) =∞∑m=0∞∑k=0A(m, k + 2m),thenF =∞∑m=0∞∑k=0(a)k+2m (b)k+2m 2−k−4m(12a + 12b + 12)k+2m (c +12)m m! k!=∞∑m=0(a)2m (b)2m 2−4m(12a + 12b + 12)2m (c +12)m m!∞∑k=0(a + 2m)k (b + 2m)k 2−k(12a + 12b + 12+ 2m)k k!(2.1)by the second equation in (1.4).The inner sum in (2.1) can be expressed as a 2F1 function in the form2F1(a + 2m, b + 2m12a + 12b + 12+ 2m; 12)=Γ(12) Γ(12a + 12b + 12)Γ(12a + 12) Γ(12b + 12)(12a + 12b + 12)2m(12a + 12)m (12b + 12)mwhich has been summed by Gauss’ second theorem in (1.3). Substitution ofthis summation into (2.1), combined with use of the first equation in (1.4),then yieldsF =Γ(12) Γ(12a + 12b + 12)Γ(12a + 12) Γ(12b + 12)∞∑k=0(12a)m (12b)m(c + 12)m m!.This last sum can be summed by Gauss’ first theorem in (1.2) when Re(2c −a − b) > −1, and the desired result in (1.1) follows. This completes the proofof Watson’s theorem.164 A. K. Rathie and R. B. ParisReferences[1] M. Abramowitz, I. Stegun (Eds.), Handbook of Mathematical Functions .Dover, New York, 1965.[2] G. E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge Univer-sity Press, Cambridge 1999.[3] R. C. Bhatt, Another proof of Watson’s theorem for summing 3F2(1), J.London Math. Soc. 40 (1965), 47–48.[4] T. M. MacRobert, Functions of a complex variable, 5th edition, Macmil-lan, London, 1962.[5] E. D. Rainville, Special Functions, Macmillan, New York, 1960.[6] L. J. Slater, Generalized Hypergeometric Functions, Cambridge UniversityPress, Cambridge, 1966.[7] G. N. Watson, A note on generalized hypergeometric series, Proc. LondonMath. Soc. (2), 23 (1925), xiii-xv.[8] F. J. W. Whipple, A group of generalized hypergeometric series: relationsbetween 120 allied series of the type F (a, b, c; e, f), Proc. London Math.Soc. (2), 23 (1925), 104–114.Received: October 3, 2008

A new proof of Watson's theorem for the series 3F2(1)

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A new proof of Watson's theorem for the series 3F2(1)

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