On arithmetic nature of $q$-analogue of the generalized Stieltjes constants
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arXiv
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| Format: | Preprint |
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2024
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| author | Chatterjee, Tapas Garg, Sonam |
| author_facet | Chatterjee, Tapas Garg, Sonam |
| contents | In this article, our aim is to extend the research conducted by Kurokawa and Wakayama in 2003, particularly focusing on the $q$-analogue of the Hurwitz zeta function. Our specific emphasis lies in exploring the coefficients in the Laurent series expansion of a $q$-analogue of the Hurwitz zeta function around $s=1$. We establish the closed-form expressions for the first two coefficients in the Laurent series of the $q$-Hurwitz zeta function. Additionally, utilizing the reflection formula for the digamma function and the identity of Bernoulli polynomials, we explore transcendence results related to $γ_0(q,x)$ for $q>1$ and $0 < x <1$, where $γ_0(q,x)$ is the constant term which appears in the Laurent series expansion of $q$-Hurwitz zeta function around $s=1$.
Furthermore, we put forth a conjecture about the linear independence of special values of $γ_0(q,x)$ along with $1$ at rational arguments with co-prime conditions, over the field of rational numbers. Finally, we show that at least one more than half of the numbers are linearly independent over the field of rationals. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_08025 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On arithmetic nature of $q$-analogue of the generalized Stieltjes constants Chatterjee, Tapas Garg, Sonam Number Theory In this article, our aim is to extend the research conducted by Kurokawa and Wakayama in 2003, particularly focusing on the $q$-analogue of the Hurwitz zeta function. Our specific emphasis lies in exploring the coefficients in the Laurent series expansion of a $q$-analogue of the Hurwitz zeta function around $s=1$. We establish the closed-form expressions for the first two coefficients in the Laurent series of the $q$-Hurwitz zeta function. Additionally, utilizing the reflection formula for the digamma function and the identity of Bernoulli polynomials, we explore transcendence results related to $γ_0(q,x)$ for $q>1$ and $0 < x <1$, where $γ_0(q,x)$ is the constant term which appears in the Laurent series expansion of $q$-Hurwitz zeta function around $s=1$. Furthermore, we put forth a conjecture about the linear independence of special values of $γ_0(q,x)$ along with $1$ at rational arguments with co-prime conditions, over the field of rational numbers. Finally, we show that at least one more than half of the numbers are linearly independent over the field of rationals. |
| title | On arithmetic nature of $q$-analogue of the generalized Stieltjes constants |
| topic | Number Theory |
| url | https://arxiv.org/abs/2404.08025 |