Related papers: A Method for Obtaining Generating Function for Cen…
The powers of generating functions and its properties are analyzed. A new class of functions is introduced, based on the application of compositions of an integer $n$, called composita. The methods for obtaining reciprocal and reverse…
For $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the $n$th generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. In particular, $T_n=T_n(1,1)$ is the central…
A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained,…
For any $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Let $p$ be an odd prime. In this paper, we…
A generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$ with $b,c\in\mathbb Z$. In this paper we investigate congruences and series for sums of terms related to central binomial…
We construct the generating function for products of inverse central binomial coefficients with harmonic numbers.
This paper presents a new generating function for Hermite polynomials of one variable in the form of $g(x,t)=\sum_{n=0}^{\infty }t^{n}H^{e}_{n}(x)$ and reveals its connection with incomplete gamma function.
For loop integrals, the reduction is the standard method. Having an efficient way to find reduction coefficients is an important topic in scattering amplitudes. In this paper, we present the generation functions of reduction coefficients…
For integers $b$ and $c$ the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Those $T_n=T_n(1,1)\ (n=0,1,2,\ldots)$ are the usual central trinomial coefficients, and…
Let $m,k$ be fixed positive integers. Determining the generating function for the number of tilings of an $m\times n$ rectangle by $k\times 1$ rectangles is a long-standing open problem to which the answer is only known in certain special…
Using the notion of the composita, we obtain a method of solving iterative functional equations of the form $A^{2^n}(x)=F(x)$, where $F(x)=\sum_{n>0} f(n)x^n$, $f(1)\neq 0$. We prove that if $F(x)=\sum_{n>0} f(n)x^n$ has integer…
It is demonstrated how an explicit expression of the (partial) sum of Tetranacci numbers can be found and proved using generating functions and the Hadamard product. We also provide a Binet-type formula for generalized Fibonacci numbers, by…
In this paper we study the coefficients of the powers of an ordinary generating function and their properties. A new class of functions based on compositions of an integer $n$ is introduced and is termed composita. We present theorems about…
Fix a prime N, and consider the action of the Hecke operator T_N on the space M_k(SL(2,Z)) of modular forms of full level and varying weight k. The coefficients of the matrix of T_N with respect to the basis {E_4^i E_6^j | 4i + 6j = k} for…
We consider the tiling generating functions of semi-hexagons and quartered hexagons with dents on their sides. In general, there are no simple product formulas for these generating functions. However, we show that the modification in the…
The exponential generating function of ordinary generating functions of diagonal sequences of general Sheffer triangles is computed by an application of Lagrange's theorem. For the special Jabotinsky type this is already known. An analogous…
Generating functions for Clebsch-Gordan coefficients of osp(1|2) are derived. These coefficients are expressed as q goes to - 1 limits of the dual q-Hahn polynomials. The generating functions are obtained using two different approaches…
In a recent article a generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal expressions was a key-point to allow to give them a statistical…
Let $ p \ge 5 $ be a prime and let $ b, c \in \mathbb{Z} $. Denote by $ T_k(b,c) $ the generalized central trinomial coefficient, i.e., the coefficient of $ x^k $ in $ (x^2 + bx + c)^k $. In this paper, we establish congruences modulo $ p^3…
In this paper, we first describe how to find the generating function for the sum of the areas under generalized Dyck paths (with an arbitrary set of steps) using Motzkin paths as a motivating example. We then focus on Motzkin and Dyck…