Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2302.13776

Part 1. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{M}_{κ,μ}\left( x\right) $ function

Alexander Apelblat·Juan Luis González-Santander
View original source ↗Read source document ↗Export metadata ↓

Abstract

First derivatives of the Whittaker function $\mathrm{M}_{κ,μ}\left(x\right) $ with respect to the parameters are calculated. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Also, it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential and logarithmic functions) from the integral representation of $\mathrm{M}_{κ,μ}\left( x\right) $. These infinite sums and integrals can be expressed in closed-form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed-form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function has been derived, as well as some finite and infinite integrals containing products of algebraic, exponential, logarithmic and Bessel functions. Finally, some reduction formulas for the Whittaker functions $\mathrm{M}_{κ,μ}\left( x\right) $ and integral Whittaker functions $\mathrm{Mi}_{κ,μ}\left( x\right) $ and $\mathrm{mi}_{κ,μ}\left( x\right) $ are calculated.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexander Apelblat, Juan Luis González-Santander. 2023-02-24. Part 1. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{M}_{κ,μ}\left( x\right) $ function. https://doi.org/10.3390/axioms12040381

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications

Let $p$ be an odd prime. We prove the extension estimate $R_{S_j}^*(2\to r)\lesssim_r 1$ for every nonzero-radius sphere $S_j\subseteq\mathbb{F}_p^4$ and every $r\geq \, 34/11$, uniformly in $p$ and $j$. This improves the Stein--Tomas exponent $10/3$ established by Iosevich and Koh (2008). We also formulate a localized spherical restriction/extension conjecture that predicts the sharp dependence of the restriction norm on the size of the physical support. This conjecture implies the spherical extension estimates $R_{S_j}^*(2\to r)\lesssim_r 1$ for every $r>3$, and yields almost-every-pin distance estimates at the conjectured Erdős--Falconer exponent in four dimensions, up to an arbitrarily small power loss in the set-size hypothesis. Using the same method, we improve the bounds supplied by Fourier decay and Plancherel at intermediate support scales and derive new almost-every-pin distance estimates in $\mathbb{F}_p^4$.

math.CA↗
arXiv2026

Dimension-free estimates for discrete maximal functions over cubes in $\mathbb Z^d$

In this short note, we establish dimension-free $\ell^p(\mathbb Z^d)$ bounds, for all $p\in(1,\infty]$, for the discrete Hardy--Littlewood maximal functions associated with cubes in $\mathbb Z^d$, answering a question that had been open for a while. The key idea is to prove dimension-free bounds for the $\ell^p(\mathbb Z^d)$ norms of the differences of the corresponding averages. This follows from an ad hoc interpretation of the associated discrete multipliers as a special continuous family of multipliers to which basic fractional integration and complex interpolation can be applied. The same method also yields an elementary proof of Bourgain's dimension-free $L^p(\mathbb R^d)$ bounds for the Hardy--Littlewood maximal function associated with cubes in $\mathbb R^d$.

math.CA↗
arXiv2026

Establishing the Polynomial Wolff Axioms for $δ$-Separated $δ$-Tubes With #o-minimality

We establish the full version of a conjecture of Guth and Zahl, giving a lower bound for the volume of a semialgebraic set that has a large intersection with a collection of $δ$-separated $δ$-tubes. Our proof uses o-minimal methods to simplify the proof of Katz and Rogers, who proved the conjecture up to a small factor. We also establish that the constants depend polynomially on the complexity of the semialgebraic set, and more generally in the #o-minimal setting.

math.CA↗