Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers
Sign Up to like & getrecommendations! Published in 2019 at "Journal of Mathematical Analysis and Applications"
DOI: 10.1016/j.jmaa.2019.05.015
Abstract: The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized… read more here.
Keywords: powers binomial; binomial coefficients; sums involving; new families ... See more keywords
Factors of sums involving q-binomial coefficients and powers of q-integers
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DOI: 10.1080/10236198.2017.1355366
Abstract: Abstract We show that, for all positive integers , , and any non-negative integers j and r with , the expression is a Laurent polynomial in q with integer coefficients, where and . This gives… read more here.
Keywords: coefficients powers; sums involving; factors sums; involving binomial ... See more keywords
Elementary proof of congruences involving sum of binomial coefficients
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DOI: 10.1142/s1793042118500938
Abstract: We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients. read more here.
Keywords: congruences involving; proof congruences; elementary proof; involving sum ... See more keywords
Supercongruences for some lacunary sums of powers of binomial coefficients
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DOI: 10.1142/s1793042118501294
Abstract: Let a,n,m ∈ ℤ+ with (a,p) = 1 and p be an odd prime. We find a supercongruence for ∑p|kapn km and related sums of powers of binomial coefficients. These results complement prior results for… read more here.
Keywords: powers binomial; sums powers; lacunary sums; binomial coefficients ... See more keywords
Factors of Sums and Alternating Sums of Products of $q$-binomial Coefficients and Powers of $q$-integers
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DOI: 10.11650/tjm/180601
Abstract: We prove that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and non-negative integers $j$ and $r$ with $j\leqslant m$, the following two expressions \begin{align*} &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{-1}\sum_{k=0}^{n_1} q^{j(k^2+k)-(2r+1)k}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack n_i-k},\\[5pt] &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{-1}\sum_{k=0}^{n_1}(-1)^k q^{{k\choose 2}+j(k^2+k)-2rk}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack… read more here.
Keywords: binomial coefficients; sums alternating; powers integers; alternating sums ... See more keywords
Degenerate binomial coefficients and degenerate hypergeometric functions
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DOI: 10.1186/s13662-020-02575-3
Abstract: In this paper, we investigate degenerate versions of the generalized p th order Franel numbers which are certain finite sums involving powers of binomial coefficients. In more detail, we introduce degenerate generalized hypergeometric functions and… read more here.
Keywords: degenerate hypergeometric; hypergeometric functions; binomial coefficients; numbers order ... See more keywords
On a Surface Associated with Pascal's Triangle
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DOI: 10.3390/sym14020411
Abstract: An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients),… read more here.
Keywords: triangle; binomial coefficients; pascal triangle; associated pascal ... See more keywords