Related papers: The solutions of four $q$-functional equations
In this note we describe solutions of the equation: $F(A(z))=G(B(z)),$ where $A,B$ are polynomials and $F,G$ are continuous functions on the Riemann sphere.
We prove a formula for the nth power of the q-derivative operator at x=0 for every function whose nth derivative at x=0 exists. We give a proof in both the real variable and the complex variable case.
We provide a general product formula for the solution of nonautonomous abstract delay equations. After having shown the convergence we obtain estimates on the order of convergence for differentiable history functions. Finally, the…
This manuscript reviews theoretical results and applications related to quadratic forms in Gaussian random variables. It summarizes definitions, canonical representations, exact and approximate distributional results, numerical inversion…
The functional equation f(p(z))=g(q(z)) is studied, where p,q are polynomials and f,g are trancendental meromorphic functions in C. We find all the pairs p,q for which there exist nonconstant f,g satisfying our equation and there exist no…
We present some completely monotonic functions involving the $q$-gamma function that are inspired by their analogues involving the gamma function.
We classify solutions of finite Morse index of the fractional Lane- Emden equation
We study and partially classify cubic rational expressions $g(x)/h(x)$ over a finite field $\mathbb{F}_q$, up to pre- and post-composition with independent M\"obius transformations. In particular, we obtain a full classification when $q$ is…
In this paper, we establish a q-analog of partial fraction decomposition formula. By using formula, we develop new closed form representations of sums of q-harmonic numbers and reciprocal q-binomial coefficients. Moreover, we give explicit…
Several kinds of q-orthogonal polynomials with |q|=1 are constructed as the main parts of the eigenfunctions of new solvable discrete quantum mechanical systems. Their orthogonality weight functions consist of quantum dilogarithm functions,…
This note reports on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type, Laguerre-type,…
In this work, the Lambert-Tsallis Wq function is used to provide analytical solutions of fractional polynomials of the type ax^r+bx^s+c = 0. This class of fractional polynomial appears in several areas of physics as well it is in the heart…
We use analytic and numerical methods to obtain the solution of the Q-ball equation of motion. In particular, we show that the profile function of the three-dimensional Q-ball can be accurately approximated by the symmetrized Woods-Saxon…
We look at the number of solutions of an equation of the form f_1*f_2*...*f_k=a in a finite field, where each f_i is a multilinear polynomial. We use two methods to construct a solution of this problem for the cases a=0, a<>0, and we…
Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…
In this paper are examined general classes of linear and non-linear analytical systems of partial differential equations. Indeed the integrability conditions are found and if they are satisfied, the solutions are given as functional series…
A four-term recurrence relation for squared spherical Bessel functions is shown to yield closed-form expressions for several types of finite weighted sums of these functions. The resulting sum rules, which may contain an arbitrarily large…
We give an explicit determinant formula for a class of rational solutions of a q-analogue of the Painlev\'e V equation. The entries of the determinant are given by the continuous q-Laguerre polynomials.
In investigation of boundary-value problems for certain partial differential equations arising in applied mathematics, we often need to study the solution of system of partial differential equations satisfied by hypergeometric functions and…
We show that a positive proportion of linear forms in four variables admit a solution in the primes that is as small as one would heuristically expect. Out of the linear forms that satisfy certain local solvability conditions, almost all…