Raising Series to A Power

S Sriram; A David Christopher

doi:10.17485/IJST/v17sp1.249

Article

VIEWS 95
PDF 38

Indian Journal of Science and Technology

DOI: 10.17485/IJST/v17sp1.249

Year: 2024, Volume: 17, Issue: Special Issue 1, Pages: 124-135

Original Article

Raising Series to A Power

S Sriram^1*, A David Christopher¹

¹PG & Research Department of Mathematics, National College, affiliated to Bharathidasan University, Trichy, Tamil Nadu, India

*Corresponding Author
Email: [email protected]

Received Date:29 August 2023, Accepted Date:05 March 2024, Published Date:31 May 2024

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

Objective. Let be a formal power series and let . In this note, we consider the function . We find that if has a series expansion at , then its coefficients are polynomials in . The coefficients of these polynomials were found to be a weighted composition sum. Methods. The method to arrive at this representation involves logarithmic derivative and exponential representation. Findings. As a consequence of this, new identities involving partition functions and binomial coefficients were obtained. Further, a particular class of Dirichlet series is found to have the form of an exponential function. Consequently, identities involving Riemann zeta function values were obtained. Novelty. The present work generalizes a class of functions considered by D’Arcais. Divisor-sum identities involving partition functions and exponential representation of Dirichlet series of this article were new to the literature.

Keywords: Polynomials, Partitions, Dirichlet Series, DivisorSum, Power Series

References

Mordell LJ. Note on Certain Modular Relations Considered by Messrs. Ramanujan, Darling, and Rogers. Proceedings of the London Mathematical Society. 1922;s2-20(1):408–416. Available from: https://dx.doi.org/10.1112/plms/s2-20.1.408
Heim B, Neuhauser M, Weisse A. Records on the vanishing of Fourier coefficients of powers of the Dedekind eta function. Research in Number Theory. 2018;4(3). Available from: https://dx.doi.org/10.1007/s40993-018-0125-y
Zouareg Y, Benoumhani M. On a conjecture of Heim and Neuhauser on some polynomials arising from modular forms and related to Fibonacci polynomials. The Ramanujan Journal. 2022;58(1):183–201. Available from: https://dx.doi.org/10.1007/s11139-021-00515-7
Heim B, Neuhauser M. The Dedekind eta function and D’Arcais-type polynomials. Research in the Mathematical Sciences. 2020;7(1):3. Available from: https://dx.doi.org/10.1007/s40687-019-0201-5
Heim B, Neuhauser M. Estimate for the largest zeros of the D’Arcais polynomials. Research in the Mathematical Sciences. 2024;11(1):1. Available from: https://dx.doi.org/10.1007/s40687-023-00412-z
Heim B, Neuhauser M, Tröger R. Zeros of recursively defined polynomials. Journal of Difference Equations and Applications. 2020;26(4):510–531. doi: 10.1080/10236198.2020.1748022
Heim B, Neuhauser M, Tröger R. Zeros transfer for recursively defined polynomials. Research in Number Theory. 2023;9(4). Available from: https://dx.doi.org/10.1007/s40993-023-00480-8
Abinash S. On 3-divisibility of 9- and 27-regular partitions. The Ramanujan Journal. 2022;57(3):1193–1207. Available from: https://dx.doi.org/10.1007/s11139-021-00463-2
Cherubini G, Mercuri P. Parity of the 8-regular partition function. The Ramanujan Journal. 2024;63(3):715–722. Available from: https://dx.doi.org/10.1007/s11139-023-00784-4
Li BQ, Steuding J. Fixed points of the riemann zeta function and dirichlet series. Monatshefte für Mathematik. 2022;198(3):581–589. Available from: https://dx.doi.org/10.1007/s00605-022-01709-x
Navas LM, Ruiz FJ, Varona JL. A connection between power series and Dirichlet series. Journal of Mathematical Analysis and Applications. 2021;493(2):124541. Available from: https://dx.doi.org/10.1016/j.jmaa.2020.124541
Chavan P, Chavan S, Vignat C, Wakhare T. Dirichlet series under standard convolutions: variations on Ramanujan’s identity for odd zeta values. The Ramanujan Journal. 2022;59(4):1245–1285. Available from: https://dx.doi.org/10.1007/s11139-022-00624-x

Copyright

© 2024 Sriram & Christopher. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Published By Indian Society for Education and Environment (iSee)