Some \(q\)-series identities related to divisor functions
type: Item
instance of: scholarly article
title: Some \(q\)-series identities related to divisor functions
zbMATH Open document ID: 0834.05005
DOI: 10.1016/0012-365X(95)00092-B
author: Karl Dilcher
published in: Discrete Mathematics
publication date: 1995-10-29T00:00:00Z
review text: \textit{K. Uchimura} [Discrete Appl. Math. 18, 73-81 (1987; Zbl 0629.10005)] studied the \(q\)-series \[ U_m(q)= \sum^\infty_{n= 1} n^m q^n \prod^\infty_{j= n+ 1} (1- q^j). \] The author proves that the generating functions of the divisor functions \(\sigma_k(n)= \sum_{d|n} d^k\) are expressed as sums of products of \(U_m(q)\) \((m= 1,\dots, k+ 1)\) and vice versa. It is also shown that the \(U_m(q)\) are closely related to certain recurrences of polynomials. Other related \(q\)-series identities are derived, including a class of binomial sums connected with the harmonic numbers.
reviewed by: Indulis Strazdins
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 11B73
zbMATH DE Number: 810257
zbMATH Keywords: \(q\)-analogue
zbMATH Keywords: \(q\)-series
zbMATH Keywords: generating functions
zbMATH Keywords: divisor functions
zbMATH Keywords: binomial sums
MaRDI profile type: Publication
cites work: Q4146776
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description: scientific article; zbMATH DE number 810257