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A PROOF OF HIRSCHHORN’S CONJECTURE ON $2^n$-DISSECTION OF EULER’S PRODUCT

Part of: Combinatorics Sequences and sets

Published online by Cambridge University Press:  08 October 2024

BIPUL KUMAR SARMAH Open the ORCID record for BIPUL KUMAR SARMAH [Opens in a new window]  and
SATYAJIT GAYAN Open the ORCID record for SATYAJIT GAYAN [Opens in a new window]
BIPUL KUMAR SARMAH*
Affiliation:
Department of Mathematical Sciences, Tezpur University, Napaam 784028, Sonitpur, Assam, India
SATYAJIT GAYAN
Affiliation:
Department of Mathematical Sciences, Tezpur University, Napaam 784028, Sonitpur, Assam, India e-mail: satyajit@tezu.ernet.in
*
e-mail: bipul@tezu.ernet.in
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Abstract

Using properties of Ramanujan’s theta functions, we give an elementary proof of Hirschhorn’s conjecture on $2^n$-dissection of Euler’s product $E(q):=(q;q)_\infty $.

Keywords

Euler’s productdissectionRamanujan’s theta functions

MSC classification

Primary: 05A30: $q$-calculus and related topics
Secondary: 11B65: Binomial coefficients; factorials; $q$-identities

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 2 , April 2025 , pp. 186 - 196
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

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