Related papers: Sums of squares of Tetranacci numbers: A generatin…
Louis W. Shapiro gave a combinatorial proof of a bilinear generating function for Chebyshev polynomials equivalent to the formula 1/(1-ax-x^2) * 1/(1-bx-x^2) = (1-x^2)/(1-abx-(2+a^2+b^2)x^2 -abx^3+x^4), where * denotes the Hadamard product.…
Convolutions for Tribonacci numbers involving binomial coefficients are treated with ordinary generating functions and the diagonalization method of Hautus and Klarner. In this way, the relevant generating function can be established, which…
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…
In this paper we evaluate sums and integrals of products of Fubini polynomials and have new explicit formulas for Fubini polynomials and numbers. As a consequence of these results new explicit formulas for p-Bernoulli numbers and…
In this paper we study the sum $$\sum_{j_1+j_2+...+j_d=n}\prod_{i=1}^d F_{k\cdot j_i},$$ where $d\geq2$ and $k\geq1$.
We state and prove product formulae for several generating functions for sequences $(a_n)_{n\ge0}$ that are defined by the property that $Pa_n+b^2$ is a square, where $P$ and $b$ are given integers. In particular, we prove corresponding…
We first summarize joint work on several preliminary canonical Lambert series factorization theorems. Within this article we establish new analogs to these original factorization theorems which characterize two specific primary cases of the…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known, however sums…
Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients and also all the sequences with…
In this article, we give a formula for the generalization of the binomial coefficient to the complex numbers as a linear combination of $\sinc$ functions. We then give a general formula to compute the integral on the real line of the…
Fibonacci sequence, generated by summing the preceding two terms, is a classical sequence renowned for its elegant properties. In this paper, leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of…
In this paper, by using bi-periodic Fibonacci numbers, we introduce the bi-periodic Fibonacci octonions. After that, we derive the generating function of these octonions as well as investigated some properties over them. Also, as another…
Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as sub-series and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the…
We establish new operational formulae of Burchnall type for the complex disk polynomials (generalized Zernike polynomials). We then use them to derive some interesting identities involving these polynomials. In particular, we establish…
We obtain explicit factored closed-form expressions for Fibonacci and Lucas sums of the form \mbox{$\sum_{k = 1}^n {F_{2rk}^3 }$} and \mbox{$\sum_{k = 1}^n {L_{2rk}^3 }$}, where $r$~and~$n$ are integers.
The idea of generating integrals analogous to generating functions is first introduced in this paper. A new proof of the well-known Finite Harmonic Series Theorem in Analysis and Analytical Number Theory is then obtained by the method of…
In this study, we introduce the generalized Tribonacci hyperbolic spinors and properties of this new special numbers system by the generalized Tribonacci numbers, which are one of the most general form of the third-order recurrence…
We investigate fractional sums of arithmetic functions over products of two or three integers, with emphasis on fixed greatest common divisors and multiplicative weights. Let $f$ be an arithmetic function satisfying $f(n) \ll n^\alpha$ for…
MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given,…