Ramanujan's1ψ1summation formula may be stated as1ψ1(a;b;q,z):=∑n=∞∞(a;q)n(b;q)nzn=(q,b/a,az,q/az;q)∞(b,q/a,z,b/az;q)∞(7.1)Note that this formula is a common extension of both theq-binomial theorem (3.1) and the Jacobi triple product identity (6.1) — see ...
Chua, K.S.: Circular summation of the 13th powers of Ramanujan’s theta function. Ramanujan J. 5, 353–354 (2001) Article MathSciNet Google Scholar Liu, X.-F., Luo, Q.-M.: A note for alternating Ramanujan’s circular summation formula. Forum Math. 27, 3189–3203 (2015) MathSci...
H.H. Chan, Z.-G. Liu, Analogues of Jacobi’s inversion formula for the incomplete elliptic integral of the first kind. Adv. Math.174, 69–88 (2003) ArticleMathSciNetMATHGoogle Scholar H.H. Chan, Z.-G. Liu, S.T. Ng, Circular summation of theta functions in Ramanujan’s lost note...
By virtue of this method, some new and elementary proofs for four partial theta function identities due to Alladi and Berkovich (2004) [8], Andrews and Warnaar (2007) [4], Berkovich [5] and Warnaar (2003) [2], as well as for Ramanujan's 1ψ11ψ1 summation formula are obtained....
we find that the dual relation of Dougall's summation theorem for the well--poised $_7F_6$-series can be utilized to construct numerous interesting Ramanujan--like infinite series expressions for $pi^{pm1}$ and $pi^{pm2}$, including an elegant formula of $pi^{-2}$ due to Guillera....
thedualsummationformulaearealso given. KeyWords:Rogers-Ramanujantypeidentities;q-series transformation formulae; Bailey’Stransform;Bailey’SLemnm;Baileypairs;generatingfunctions;Carlitz inversions;Carlitz inversioochains;Abel’8 Lemma 独创性说明 作者郑重声明:本博士学位论文是作者本人在导师指导下进 行的研究工...
where q and n are in N and the summation is over a reduced residue system mod q. Many properties were derived in [13] and elaborated in [8]. Cohen [3] generalized this arithmetical function in the following way. Definition 1.2. Let β∈ N. The c (β) ...
The mathematician Ramanujan introduced a summation in 1918, now known as the Ramanujan sum cq(n). In a companion paper (Part I), properties of Ramanujan su... Vaidyanathan,P P. - 《Signal Processing IEEE Transactions on》 被引量: 30发表: 2014年 ...
Four classes of quartic theta hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformations are proved that express the quartic series in terms of well-poised, quadratic and cubic ones. Thirty new summation formulae for terminating quartic ...
3 Ramanujan’s Proof of the q-Gauss Summation Theorem . . . . . 10 1. 4 Corollaries of (1. 2. 1) and (1. 2. 5) . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 5 Corollaries of (1. 2. 6) and (1. 2. 7) . . . . . . . . ....