Related papers: Confluent hypergeometric expansions of the conflue…
Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the…
Inspired by certain interesting recent extensions of the gamma, beta and hypergeometric matrix functions, we introduce here new extension of the gamma and beta matrix function. We also introduce new extensions of the Gauss hypergeometric…
We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinite summations…
We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schr\"odinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse…
Hypergeometric sequences obey first-order linear recurrence relations with polynomial coefficients and are commonplace throughout the mathematical and computational sciences. For certain classes of hypergeometric sequences, we prove linear…
Two linear recurrences exhibit mirror symmetry connecting the constants $e$ and $\pi$. When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational…
Given the growing quantity of proposals and works of basic hypergeometric functions in the scope of $q$-calculus, it is important to introduce a systematic classification of $q$-calculus. Our aim in this article is to investigate certain…
Gauss's arithmetic-geometric mean (AGM) which is described by two variables iteration $(a_n, b_n)\rightarrow (a_{n+1}, b_{n+1})$ by $a_{n+1}=(a_n+b_n)/2,\ b_{n+1}=\sqrt{a_nb_n}$. We extend it to three variables iteration $(a_n, b_n,…
Hypergeometric sequences are rational-valued sequences that satisfy first-order linear recurrence relations with polynomial coefficients; that is, $\langle u_n \rangle_{n=0}^\infty$ is hypergeometric if it satisfies a first-order linear…
We study a family of double confluent Heun equations of the form $\mathcal E=0$, where $\mathcal L=\mathcal L_{\lambda,\mu,n}$ is a family of differential operators of order two. They depend on complex parameters $\lambda$, $\mu$, $n$. Its…
In this paper, we use some standard numerical techniques to approximate the hypergeometric function $$ {}_2F_1[a,b;c;x]=1+\frac{ab}{c}x+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{x^2}{2!}+\cdots $$ for a range of parameter triples $(a,b,c)$ on the…
In this work we establish new forms of Heun-to-Heun transformations and Heun-to-Hypergeometric transformations. The transformations are realised by changing the independent variable in a non-linear way. Using these we also point out some…
Inspired by the paper of Bonichon, Bousquet-M\'elou, Dorbec and Pennarun, we give a system of functional equations which characterise the ordinary generating function, $U(x),$ for the number of planar Eulerian orientations counted by edges.…
By using various expansions of the parametric digamma function and the method of residue computations, we study three variants of the linear Euler sums, related Hoffman's double $t$-values and Kaneko-Tsumura's double $T$-values, and…
Sufficient conditions are obtained on the parameters of Lommel function of the first kind, generalized Struve function of the first kind and the confluent hypergeometric function under which these special functions become exponential convex…
We prove some extension theorems for quaternionic holomorphic functions in the sense of Fueter. Starting from the existence theorem for the nonhomogeneous Cauchy-Riemann-Fueter Problem, we prove that an $\mathbb{H}$-valued function $f$ on a…
The Leaver solutions in series of Coulomb wave functions for the confluent Heun equation (CHE) are given by two-sided infinite series, that is, by series where the summation index $n$ runs from minus to plus infinity [E. W. Leaver, J. Math.…
After a brief introduction to Heun type functions we note that the actual solutions of the eigenvalue equation emerging in the calculation of the one loop contribution to QCD from the Belavin-Polyakov-Schwarz-Tyupkin instanton and the…
We fix three natural numbers $k, n, N$, such that $n+k+1=N$, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of $N$ hyperplanes in a $k$-dimensional affine space, the other is an…
It is shown that Schr\"odinger equation with combination of three potentials U = - {\alpha} r^{-1} + {\beta} r + kr^{2}, Coulomb, linear and harmonic, the potential often used to describe quarkonium, is reduced to a bi-confluent Heun…