Related papers: On the $q$-partial differential equations and $q$-…
A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…
For $\delta$ an $m$-tuple of analytic functions, we define an algebra $\hidg$, contained in the bounded analytic functions on the analytic polyhedron $ {|\delta^l(z)| < 1, \ 1 \leq l \leq m}$, and prove a representation formula for it. We…
$q$-Analogues of the coefficients of $x^a$ in the expansion of $\prod_{j=1}^N (1+x+...+x^j)^{L_j}$ are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the ``$q$-supernomial coefficients'' are…
We give an overview about the power product expansion of the exponential series and derive some q-analogs
For a simple connected graph $G$, the $Q$-generating function of the numbers $N_k$ of semi-edge walks of length $k$ in $G$ is defined by $W_Q(t)=\sum\nolimits_{k = 0}^\infty {N_k t^k }$. This paper reveals that the $Q$-generating function…
n this paper, we present $q$-Bernoulli and $q$-Euler polynomials generated by the third Jackson $q$-Bessel function to construct new types of $q$-Lidstone expansion theorem. We prove that the entire function may be expanded in terms of…
We introduce a new family of special functions, namely $q$-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to $q$-analogues of Poisson distributions. We focus our attention on their structural…
We propose a new definition of the q-exponential function. Our q-exponential function maps the imaginary axis into the unit circle and the resulting q-trigonometric functions are bounded and satisfy the Pythagorean identity.
In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions.…
This is the second part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. In this part, we give constructive results generalizing previous ones obtained by the author in…
In this paper, we intend to revisit Theorem 2 of [3] formulating it in a way that, weakening the hypotheses and, at the same time, highlighting the richer conclusion allowed by the proof, it can potentially be applicable to a broader range…
Using a general $q$-summation formula, we derive a generating function for the $q$-Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the $q$-Hahn polynomials. A new proof of the orthogonality…
We present a deformed algebra related to the q-exponential and the q-logarithm functions that emerge from nonextensive statistical mechanics. We also develop a q-derivative (and consistently a q-integral) for which the q-exponential is an…
We extend some definitions and give new results about the theory of slice analysis in several quaternionic variables. The sets of slice functions which are respectively slice, slice regular and circular w.r.t. given variables are…
A simple version for the extension of the Taylor theorem to the operator functions was found. The expansion was done with respect to a value given by a diagonal matrix for the non-commutative case, and the coefficients are given both by…
We obtain q-analogues of the Sylvester, Ces\`aro, Pasternack, and Bateman polynomials. We also derive generating functions for these polynomials.
A general scheme for tridiagonalising differential, difference or q-difference operators using orthogonal polynomials is described. From the tridiagonal form the spectral decomposition can be described in terms of the orthogonality measure…
We present an extension theorem for a separately holomorphic function which is polynomial/rational in some variables.
A $q$-analogue of the multiple gamma functions is introduced, and is shown to satisfy the generalized Bohr-Morellup theorem. Furthermore we give some expressions of these function.
In this paper, we propose some algorithms for analytical solution construction to nonlinear polynomial partial differential equations with constant function coefficients. These schemes are based on one-(single), two- (double) or three-…