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A series transformation formula and related polynomials

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  • Khristo N. Boyadzhiev

Abstract

We present a formula that turns power series into series of functions. This formula serves two purposes: first, it helps to evaluate some power series in a closed form; second, it transforms certain power series into asymptotic series. For example, we find the asymptotic expansions for λ > 0 of the incomplete gamma function γ ( λ , x ) and of the Lerch transcendent Φ ( x , s , λ ) . In one particular case, our formula reduces to a series transformation formula which appears in the works of Ramanujan and is related to the exponential (or Bell) polynomials. Another particular case, based on the geometric series, gives rise to a new class of polynomials called geometric polynomials.

Suggested Citation

  • Khristo N. Boyadzhiev, 2005. "A series transformation formula and related polynomials," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2005, pages 1-18, January.
    • Handle: RePEc:hin:jijmms:792107
    DOI: 10.1155/IJMMS.2005.3849
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    Cited by:

    1. Manuel D. Ortigueira, 2022. "A New Series Representation and the Laplace Transform for the Lognormal Distribution," Mathematics, MDPI, vol. 10(19), pages 1-13, September.
    2. Sunil Kumar Sharma & Waseem Ahmad Khan & Cheon-Seoung Ryoo & Ugur Duran, 2022. "Diverse Properties and Approximate Roots for a Novel Kinds of the ( p , q )-Cosine and ( p , q )-Sine Geometric Polynomials," Mathematics, MDPI, vol. 10(15), pages 1-18, July.
    3. Hacène Belbachir & Yahia Djemmada, 2020. "Generalized Geometric Polynomials Via Steffensen’s Generalized Factorials and Tanny’s Operators," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(4), pages 1713-1727, December.
    4. Sunil Kumar Sharma & Waseem A. Khan & Cheon Seoung Ryoo, 2020. "A Parametric Kind of Fubini Polynomials of a Complex Variable," Mathematics, MDPI, vol. 8(4), pages 1-16, April.
    5. Khristo N. Boyadzhiev, 2011. "Series transformation formulas of Euler type, Hadamard product of series, and harmonic number identities," Indian Journal of Pure and Applied Mathematics, Springer, vol. 42(5), pages 371-386, October.

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