Departmental Bulletin Paper q シフト因子の対数に関するベキ級数展開とその解析

中井, 日佐司  ,  ナカイ, ヒサシ  ,  Nakai, Hisashi

28(1) ( 28(1) )  , pp.1 - 9 , 2016-02-01 , The University of Electro-Communications
Series expansion ln(1 ? a) ? Lq (a) is given for the logarithm of the q-shifted factorial (or q- Pochhammer symbol) defined by infinite product (a; q) := (1 ? a)(1 ? aq)(1 ? aq2 ) ? ? ? , whereLq (a) := ∑∞(aq)m /[m(1 ? qm )] within 0 ? a < 1 and 0 ? q < 1. The divergent factor 1/(1 ? q) isseparated from the expansion, and it is found out that the (a; q)∞ as a function of a is damped by thisfactor as damping coe?cient. Moreover, using this expansion, both of the upper and lower bounds of(a; q)∞ are given in terms of dilogarithms. The di?erence between both of the bounds is estimated less than 0.02.We construct a polynomial approximation on Lq , then estimate remainder term appearing in the approximation, and relate the remainder term to estimated relative error. This estimated error is compared to the error of numerical experiment in calculation on (a; q)∞ using the polynomial approx- imation. Both of the errors are consistent each other within the significant digits of the experiment.

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