Related papers: The generalized Marcum $Q-$function: an orthogonal…
Novel analytic solutions are derived for integrals that involve the generalized Marcum Q-function, exponential functions and arbitrary powers. Simple closed-form expressions are also derived for the specific cases of the generic integrals.…
Special functions have been used widely in many problems of applied sciences. However, there are considerable numbers of problems in which exact solutions could not be achieved because of undefined sums or integrals involving special…
Polynomial sequence ${P_m}_{m\geq0}$ is $q$-logarithmically concave if $P_{m}^2-P_{m+1}P_{m-1}$ is a polynomial with nonnegative coefficients for any $m\geq{1}$. We introduce an analogue of this notion for formal power series whose…
The generalized Marcum functions appear in problems of technical and scientific areas such as, for example, radar detection and communications. In mathematical statistics and probability theory these functions are called the noncentral…
We introduce a generalization $G^{(\alpha)}(X)$ of the truncated logarithm $\mathcal{L}_1(X) = \sum_{k=1}^{p-1}X^k/k$ in characteristic $p$, which depends on a parameter $\alpha$. The main motivation of this study is $G^{(\alpha)}(X)$ being…
The paper studies logarithmic convexity and concavity of power series with coefficients involving q-gamma functions or q-shifted factorials with respect to a parameter contained in their arguments. The principal motivating examples of such…
We study a q-generalization of the classical Laguerre/Hermite orthogonal polynomials. Explicit results include: the recursive coefficients, matrix elements of generators for the Heisenberg algebra, and the Hankel determinants. The power of…
Methods and an algorithm for computing the generalized Marcum $Q-$function ($Q_{\mu}(x,y)$) and the complementary function ($P_{\mu}(x,y)$) are described. These functions appear in problems of different technical and scientific areas such…
Fractional $q$-extensions of some classical $q$-orthogonal polynomials are introduced and some of the main properties of the new defined functions are given. Next, a fractional $q$-difference equation of Gauss type is introduced and solved…
Expressions for the summation of the series involving the Laguerre polynomials \[S_m(\pm\nu, \pm p)\equiv e^{-x}\sum_{n=0}^\infty \frac{x^n\,L_n^{(\nu)}(x)}{(1\pm \nu\pm p)_n}\frac{(f+m)_n}{(f)_n}\] for any non-negative integers $m$ and $p$…
Expressions for the summation of a new series involving the Laguerre polynomials are obtained in terms of generalized hypergeometric functions. These results provide alternative, and in some cases simpler, expressions to those recently…
We give an overview about the power product expansion of the exponential series and derive some q-analogs
In this paper, we present a new generalization of the Lucas numbers by matrix representation using Genaralized Lucas Polynomials. We give some properties of this new generalization and some relations between the generalized order-k Lucas…
Power series are introduced that are simultaneously convergent for all real and p-adic numbers. Our expansions are in some aspects similar to those of exponential, trigonometric, and hyperbolic functions. Starting from these series and…
In this paper, by use of matrix inversions, we establish a general $q$-expansion formula of arbitrary formal power series $F(z)$ with respect to the base $$\left\{z^n\frac{(az:q)_n}{(bz:q)_n}\bigg|n=0,1,2\cdots\right\}.$$ Some concrete…
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.
The purpose of this paper is to generalize this relation of symmetry between the power sum polynomials and the generalized Euler polynomials to the relation between the power sum polynomials and the generalized higher-order Euler…
There is proposed the Maillet--Malgrange type theorem for a generalized power series (having complex power exponents) formally satisfying an algebraic ordinary differential equation. The theorem describes the growth of the series…
By making use of the familiar Mathieu series and its generalizations, the authors derive a number of new integral representations and present a systematic study of probability density functions and probability distributions associated with…
In this paper we study the genralized q-Euler numbers and polynomials. From our results, we derive some interesting congruences related tothe generalized q-Euler numbers.