Related papers: An orthogonality relation for the Whittaker functi…

Recently, Yakubovich [Opuscula Math. 26 (2006) 161--172] and Passian et al. [J. Math. Anal. Appl. doi:10.1016/j.jmaa.2009.06.067] have presented alternative proofs of an orthogonality relation obeyed by the Macdonald functions of imaginary…

First derivatives with respect to the parameters of the Whittaker function $\mathrm{W}_{\kappa ,\mu }\left( x\right) $ are calculated. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of…

Orthogonality relations for conical or Mehler functions of imaginary order are derived and expressed in terms of the Dirac delta function. This work extends recently derived orthogonality relations of associated Legendre functions.

Discrete analogs of the index Whittaker transform are introduced and investigated. It involves series and integrals with respect to a second parameter of the Whittaker function $W_{\mu, {i n} }(x), \ x >0, \ \mu \in \mathbb{R}, \ n \in…

First derivatives of the Whittaker function $\mathrm{M}_{\kappa ,\mu }\left(x\right) $ with respect to the parameters are calculated. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of…

Uniform asymptotic expansions are derived for Whittaker's confluent hypergeometric functions $M_{\kappa,\mu}(z)$ and $W_{\kappa,\mu}(z)$, as well as the numerically satisfactory companion function $W_{-\kappa,\mu}(ze^{-\pi i})$. The…

In this work we study the associated Legendre functions of the second kind with a purely imaginary argument $Q^k_\ell(\mathbb{i}\, x)$. We derive the conditions under which they provide a set of square integrable functions when $x \in…

This article considers Whittaker's function $W_{\kappa ,\mu }$ where $\kappa$ is real and $\mu$ is real or purely imaginary. Then $\varphi (x)=x^{-\mu -1/2}W_{\kappa ,\mu }(x)$ arises as the scattering function of a continuous time linear…

We construct the non-stationary Ruijsenaars functions (affine analogue of the Macdonald functions) in the special case $\kappa=t^{-1/N}$, using the intertwining operators of the Ding-Iohara-Miki algebra (DIM algebra) associated with…

In this paper, we obtain a $(p,\nu)$-extension of the Whittaker function $M_{\kappa,\mu}(z)$ by using the extended confluent hypergeometric function of the first kind $\Phi_{p,\nu}(b;c;z)$ introduced in Parmar et al. [J. Classical Anal. 11…

In this paper, we expand the theory of Weierstrassian elliptic functions by introducing auxiliary zeta functions $\zeta_\lambda$, zeta differences of first kind $\Delta_\lambda$ and second kind $\Delta_{\lambda,\mu}$ where…

Strichartz inequality for the solutions of free Schr\"odinger equation associated with Dunkl Hermite operator $H_\kappa$ is generalized to any system of orthonormal functions with initial data. A relation between the kernels of…

In his groundbreaking work on pair correlation, Montgomery analyzed the distribution of the differences $\gamma'-\gamma$ between ordinates $\gamma$ of the nontrivial zeros of the Riemann zeta function, assuming the Riemann Hypothesis. In…

Orthogonal polynomials and expansions are studied for the weight function $h_\kappa^2(x) \|x\|^{2\nu} (1-\|x\|^2)^{\mu-1/2}$ on the unit ball of $\mathbb{R}^d$, where $h_\kappa$ is a reflection invariant function, and for related weight…

We compute the second moment of the Riemann zeta function for shifted arguments over a domain that extends the ones in the literature. We use the Riemann-Siegel formula for the error term in the approximate functional equation and take the…

Following the work of the second author, a class of summation formulas attached to index transforms is studied in this paper. Our primary results concern summation and integral formulas with respect to the second index of the Whittaker…

A series of the form $\sum_{\ell=0}^{\infty}c(\kappa,\ell)\,M_{\kappa,\ell+1/2}(r_{0})W_{\kappa,\ell+1/2}(r)P_{\ell}(\cos(\gamma))$ is evaluated explicitly where $c(\kappa,\ell)$ are suitable complex coefficients, $M_{\kappa,\mu}$ and…

We build on a recent paper on Fourier expansions for the Riemann zeta function. We establish Fourier expansions for certain $L$-functions, and offer series representations involving the Whittaker function $W_{\gamma,\mu}(z)$ for the…

For $\alpha\geq 0$, $\beta<1$ and $\gamma\geq 0$, the class $\mathcal{W}_{\beta}(\alpha,\gamma)$ satisfies the condition \begin{align*} {\rm Re\,} \left( e^{i\phi}\left((1-\alpha+2\gamma)f/z+(\alpha-2\gamma)f'+ \gamma…

Multivariate orthogonal polynomials in $D$ real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials,…