Applications of degenerate q-Euler and q-Changhee polynomials with weight α

In this paper, we give new identities involving degenerate q-Euler polynomials with weight α and q-Changhee polynomials of the second kind with weight α, using the Faà di Bruno formula and some identities of the Bell polynomials of the second kind

ON BINOMIAL SUMS WITH THE TERMS OF SEQUENCES {g_kn} AND {h_kn}

In this paper, we derive sums and alternating sums of products of terms ofthe sequences $\left\{ g_{kn}\right\}$ and $\left\{ h_{kn}\right\}$ withbinomial coefficients. For example,\begin{eqnarray*} &\sum\limits_{i=0}^{n}\binom{n}{i}\left( -1\right) ^{i} \left(c^{2k}\left(-q\right) ^{k}+c^{k}v_{k}+1\right)^{-ai}h_{k\left( ai+b\right) }h_{k\left(ai+e\right) } \\ &=\left\{ \begin{array}{clc} -\Delta ^{\left( n+1\right) /2}g_{k\left( an+b+e\right) }g_{ka}^{n}\left( c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{-an} & \text{if }n\text{ is odd,} & \\ \Delta ^{n/2}h_{k\left( an+b+e\right) }g_{ka}^{n}\left( c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{-an} & \text{if }n\text{ is even,} & \end{array}% \right.\end{eqnarray*}%and\begin{eqnarray*} &&\sum\limits_{i=0}^{n}\binom{n}{i}i^{\underline{m}}g_{k\left( n-ti\right) }h_{kti} \\ &&=2^{n-m}n^{\underline{m}}g_{kn}-n^{\underline{m}}\left( c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{n\left( 1-t\right) }h_{kt}^{n-m}g_{k\left( tm+tn-n\right) },\end{eqnarray*}%where $a, b, e$ is any integer numbers, $c$ is nonzero real number and $m$is nonnegative integer

On Certain Hessenberg Matrices Related with Linear Recurrences

In this paper, we present the permanents and determinants of some Hessenbergmatrices. Also, some special cases for permanents are given

Some congruences related to harmonic numbers and the terms of the second order sequences

In this paper, with helps of some combinatorial identities, we investigate various basic congruences involving harmonic numbers and terms of the second order sequences {Ukn} and {Vkn}

Nonlinear variants of the generalized Filbert and Lilbert matrices

In this paper, we present variants of the generalized Filbert and Lilbert matrices by products of the general Fibonacci and Lucas numbers whose indices are in certain nonlinear forms of the indices with certain integer parameters. We derive explicit formulæ for inverse matrix, LU -decomposition and inverse matrices L-1 and U-1 for all matrices. Generally, we present q -versions of these matrices and their related results

New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries

In this paper, we present new analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries consist of the Fibonacci and Lucas numbers. We shall derive explicit formulæ for their LU-decompositions and inverses. To prove the claimed results, we write all the identities to be proven in q-word and then use the celebrated Zeilberger algorithm to prove required q-identities. © 2020, Hacettepe University. All rights reserved