Generalized exponential sums and the power of computers

Castro, Francis N.; González, Oscar E.; Medina, Luis A.

doi:10.2140/involve.2018.11.127

Cited by 1 publication

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References 18 publications

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“…Like this, there exist many different examples (cf. 10 [1]- [65]; see also the references cited in each of these earlier works).…”

Section: Introductionmentioning

confidence: 99%

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Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers

Şimşek

2019

Journal of Mathematical Analysis and Applications

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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 that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have been obtained and at the same time previously well-known formulas and identities have been generalized. Moreover, on this occasion, we identify new families of polynomials including some families of well-known numbers such as Bernoulli numbers, Euler numbers, Stirling numbers, Franel numbers, Catalan numbers, Changhee numbers, Daehee numbers and the others, and also for the polynomials such as the Legendre polynomials, Michael Vowe polynomial, the Mirimanoff polynomial, Golombek type polynomials, and the others. We also give both Riemann and p-adic integral representations of these polynomials. Finally, we give combinatorial interpretations of these new families of numbers, polynomials and finite sums of the powers * Corresponding author of binomial coefficients. We also give open questions for ordinary generating functions for these numbers.

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“…Like this, there exist many different examples (cf. 10 [1]- [65]; see also the references cited in each of these earlier works).…”

Section: Introductionmentioning

confidence: 99%

“…Substituting λ = −1 into (3.2), we have generalized the p-th order alterne Castro et al [10] also studied power sums of L j (n) which so-called the j-th order Franel numbers and also the j -th order alterne Franel numbers, given as follows:…”

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confidence: 99%

Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers

Şimşek

2019

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

Generalized exponential sums and the power of computers

Cited by 1 publication

References 18 publications

Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers

Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers

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