Derivation of computational formulas for certain class of finite sums: Approach to generating functions arising from p$$ p $$‐adic integrals and special functions

Şimşek, Yılmaz

doi:10.1002/mma.8321

Cited by 4 publications

(11 citation statements)

References 25 publications

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“…(see also Simsek, 2021cSimsek, , 2021d2022b, 2022c. Simsek (2022b (Equation (59))) gave the following formula 𝒚(𝑟 − 1, 𝜗) + (𝜗 − 1)𝒚(𝑟, 𝜗) = (−1) 𝑟 (𝑟 + 1)𝜗 𝑟+1 .…”

Section: Introductionmentioning

confidence: 99%

“…(for detail, see Simsek, 2021bSimsek, , 2021cSimsek, , 2021d2022b, 2022c. From the above equation, we have (Simsek, 2021b(Simsek, , 2021c2022b, 2022c.…”

Section: Introductionmentioning

confidence: 99%

“…(for detail, see Simsek, 2021bSimsek, , 2021cSimsek, , 2021d2022b, 2022c. From the above equation, we have (Simsek, 2021b(Simsek, , 2021c2022b, 2022c. Kilar and Simsek (2017) defined the following the Fubini type numbers 𝑎 𝑟 (𝑛) :…”

Section: Introductionmentioning

confidence: 99%

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Certain Finite Sums Pertaining to Leibnitz, Harmonic and Other Special Numbers

Kilar¹

2022

Gazi University Journal of Science Part A: Engineering and Innovation

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The present manuscript deals with some certain finite sums and identities pertaining to some special numbers. Using generating functions methods, some relations and identities involving the Apostol type Euler and combinatorial numbers, and also the Fubini type numbers and polynomials, are given. Then, by using some certain classes of special finite sums involving the following rational sum which is defined by Simsek (2021b): y(r,ϑ)=∑_(b=0)^r▒〖(-1)^r/((1+b) ϑ^(b+1) 〖(ϑ-1)〗^(r-b+1) ),〗many new certain finite sums and formulas related to the Leibnitz, Harmonic, Changhee, and Daehee numbers are obtained. Moreover, some applications of these results are presented.

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Section: Introductionmentioning

confidence: 99%

“…(for detail, see Simsek, 2021bSimsek, , 2021cSimsek, , 2021d2022b, 2022c. From the above equation, we have (Simsek, 2021b(Simsek, , 2021c2022b, 2022c.…”

Section: Introductionmentioning

confidence: 99%

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Certain Finite Sums Pertaining to Leibnitz, Harmonic and Other Special Numbers

Kilar¹

2022

Gazi University Journal of Science Part A: Engineering and Innovation

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“…Remark 5. Putting 𝑣 = 0, 𝜆 = 𝑝 = 1 in (6), we have (49), which is also given by generating function (32). By using (32), Charalambides 7 Exercise 30*, p. 272 also gave the following formulas:…”

mentioning

confidence: 99%

“…Here, using generating functions with their functional equations, we obtain some novel formulas for the sums of inverses of binomial coefficients derived from the numbers 𝑦 (𝑚, 𝜆). By combining (32) with ( 11) and ( 13) yields the following functional equation:…”

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confidence: 99%

Generating functions for series involving higher powers of inverse binomial coefficients and their applications

Şimşek

2023

Preprint

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The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients related to each other. A unique relation between the Euler-Frobenius polynomial and B-spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and finite sums are derived. Moreover, A few recurrence relations containing these particular finite sums are given. Using these recurence relations, we give a solution of the problem which was given by Charalambides7, Exercise 30, p. 273. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums.

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Generating functions for series involving higher powers of inverse binomial coefficients and their applications

Şimşek

2023

Math Methods in App Sciences

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The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients are related to each other. A relation between the Euler-Frobenius polynomial and B-spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and special finite sums are derived.Moreover, a few recurrence relations containing these particular finite sums are given. Using these recurrence relations, we give a solution of the problem which was given by Charalambides. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums.

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Derivation of computational formulas for certain class of finite sums: Approach to generating functions arising from p$$ p $$‐adic integrals and special functions

Cited by 4 publications

References 25 publications

Certain Finite Sums Pertaining to Leibnitz, Harmonic and Other Special Numbers

Certain Finite Sums Pertaining to Leibnitz, Harmonic and Other Special Numbers

Generating functions for series involving higher powers of inverse binomial coefficients and their applications

Generating functions for series involving higher powers of inverse binomial coefficients and their applications

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