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Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functions

Author

Listed:
  • Juan Luis González-Santander

    (Department of Mathematics, Universidad de Oviedo, 33007 Oviedo, Asturias, Spain
    These authors contributed equally to this work.)

  • Fernando Sánchez Lasheras

    (Department of Mathematics, Universidad de Oviedo, 33007 Oviedo, Asturias, Spain
    These authors contributed equally to this work.)

Abstract

We calculate some infinite sums containing the digamma function in closed form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as parameter differentiation formulas for the beta incomplete function, reduction formulas of 3 F 2 hypergeometric functions, or a definite integral which does not seem to be tabulated in the most common literature. As an application of certain sums involving the digamma function, we calculated some reduction formulas for the parameter differentiation of the Mittag–Leffler function and the Wright function.

Suggested Citation

  • Juan Luis González-Santander & Fernando Sánchez Lasheras, 2023. "Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functions," Mathematics, MDPI, vol. 11(8), pages 1-16, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1937-:d:1128129
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    References listed on IDEAS

    as
    1. Juan Luis González-Santander & Fernando Sánchez Lasheras, 2022. "Finite and Infinite Hypergeometric Sums Involving the Digamma Function," Mathematics, MDPI, vol. 10(16), pages 1-12, August.
    2. Alexander Apelblat, 2020. "Differentiation of the Mittag-Leffler Functions with Respect to Parameters in the Laplace Transform Approach," Mathematics, MDPI, vol. 8(5), pages 1-22, April.
    3. Alexander Apelblat & Juan Luis González-Santander, 2021. "The Integral Mittag-Leffler, Whittaker and Wright Functions," Mathematics, MDPI, vol. 9(24), pages 1-34, December.
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    Cited by:

    1. Juan Luis González-Santander & Fernando Sánchez Lasheras, 2023. "A Note on Some Generalized Hypergeometric Reduction Formulas," Mathematics, MDPI, vol. 11(16), pages 1-8, August.
    2. Sergei Sitnik, 2023. "Editorial for the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”," Mathematics, MDPI, vol. 11(15), pages 1-7, August.

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    1. Juan Luis González-Santander & Fernando Sánchez Lasheras, 2022. "Finite and Infinite Hypergeometric Sums Involving the Digamma Function," Mathematics, MDPI, vol. 10(16), pages 1-12, August.
    2. Alexander Apelblat & Juan Luis González-Santander, 2021. "The Integral Mittag-Leffler, Whittaker and Wright Functions," Mathematics, MDPI, vol. 9(24), pages 1-34, December.

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