The t-coefficient method III: A general series expansion for the product of theta functions with different bases and its applications

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:449

Article

Ma, Xinrong¹  Wei, Ruizhong² 

[1] Soochow Univ, Dept Math, Suzhou 215006, Peoples R China

[2] Lakehead Univ, Dept Comp Sci, Thunder Bay, ON P7B 5E1, Canada

关键词: Jacobi's triple product identity;    Bilateral series;    Cubic theta function;    t-coefficient method;    Series expansion;    Circular summation;    Schroter formula;   

DOI  :  10.1016/j.jmaa.2016.12.012

来源: Elsevier

【 摘 要 】

By means of Jacobi's triple product identity and the t-coefficient method, we establish a general series expansion formula for the product of arbitrary two theta functions with bases p and q: theta(az; p)theta(bz(t); q) = Sigma(n=-infinity) tau(p)(n)(az)(n) theta((-1/a)(t)bp(t(t+1)/2-tn); p(t2) q), from which some new results on products of arbitrary finitely many theta functions and theta identities associated with Ramanujan's circular summation can be derived, among them the most interesting ones include S-k=0(mn-1) Pi(n)(i=1) theta(-ay(i)w(k); q) = mn Gamma(0; y(1), y(2),..., y(n)) x theta(- y(1)y(2)... y(n))(m)a(mn)q(nm2/2-nm/2); q(nm2)), where omega is a primitive mn-th root of unity and Gamma(s; y(1),y(2),..., y(n)) = Sigma(i1+i2+...+in=s) q(1/2) Sigma(n)(j-1) i(j)(2) Pi(j=1) y(j)(ij). Furthermore, for y(1)y(2)y(3) = -q and omega = exp(pi i/3), Gamma(3)(0; y(1), y(2), y(3)) +q(-3/2)Gamma(3)(1; y(1), y(2), y(3)) 1/(q; q)(infinity)(3) Pi(2)(i=0)Pi(3)(j=1) theta(-y(j)omega(2i); q). The former contains Chan and Liu's circular summation (cf. Chan and Liu (2010) [12]) as a special case y(1)y(2) ... y(n) = 1. The latter generalizes both Borweins and Garvan's well-known cubic theta function identity and Schultz's bivariate-generalization (cf. Borwein et al. (1994) [9] and Schultz (2013) [28]), in which y(1) = y(2) = y(3) = omega q(1/3) and y(1) = y(3)/z(1), y(2) = y(3)/z(2), y(3)(3) = -qz(1)z(2) respectively. (c) 2016 Elsevier Inc. All rights reserved.

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