双变量部分theta函数的乘积公式与性质  

Product Formula and Allied Properties for the Bivariate Partial theta Function

作　　者：王瑾 Jin WANG(College of Mathematics and Computer Science,Zhejiang Normal University,Jinhua 321004,P.R.China)

机构地区：[1]浙江师范大学数学与计算机科学学院,金华321004

出　　处：《数学学报（中文版）》2022年第1期53-66,共14页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金(12001492,11971341);浙江省自然科学基金资助项目(LQ20A010004)。

摘　　要：本文研究一类新的双变量部分theta函数,它是经典部分theta函数的推广,主要围绕这类函数的乘积公式、递推关系、级数展开等性质展开讨论.作为主要结果,我们建立了任意两个双变量部分theta函数的乘积公式,推广了Andrews-Warnaar经典部分theta函数的乘积公式,发现了双变量部分theta函数所满足的二阶递推关系,得到了双变量部分theta函数θ(q,x;ab)关于{θ(q,axq^(n);b)|n≥0}和{θ(q,xq^(n);b)|n≥0}的级数展开式.作为这些结果的进一步应用,还给出了_(3)Φ_(2)级数的新的乘积公式和双变量部分theta函数的三元表示.The present paper is devoted to the study of the so-called bivariate partial theta function which is first introduced by the author and contains the classical partial theta function as a special case.We focus on its possible product formula,recurrence relation,and series expansion and so on.As main results,we establish a product formula of any two bivariate partial theta functions.It is a generalization of Andrews-Warnaar’s product formula for the classical partial theta functions.At the same time,we obtain a second order recurrence relation satisfied by this bivariate partial theta function.Finally,we present two series expansions of the bivariate partial theta functionθ(q,x;ab)with respect to{θ(q,axq^(n);b)|n≥0}and{θ(q,xq^(n);b)|n≥0},respectively.As further applications of these results,we also find a product formula of two _(3)Φ_(2) series and a ternary representation of the bivariate partial theta function.

关 键 词：双变量部分theta函数 乘积公式 级数展开 递推关系 t-系数法 

分 类 号：O157.1[理学—数学]