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Probabilistic Degenerate Bernoulli And Euler Polynomials Of Complex Variable

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  • WENCONG LIU

    (School of Mathematics, Northwest University, Xi’an 710119, Shaanxi, P. R. China†School of Science, Xi’an Technological University, Xi’an 710021, Shaanxi, P. R. China)

  • TAEKYUN KIM

    (��School of Science, Xi’an Technological University, Xi’an 710021, Shaanxi, P. R. China‡Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea)

  • DAE SAN KIM

    (�Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea)

  • YUANKUI MA

    (��School of Science, Xi’an Technological University, Xi’an 710021, Shaanxi, P. R. China)

Abstract

Let X be a random variable whose moment generating function exists in a neighborhood of the origin. Recently, the degenerate Euler (Bernoulli) polynomials of complex variable have been studied. In this paper, as probabilistic extensions of those polynomials, we study the probabilistic degenerate Euler (Bernoulli) polynomials of complex variable associated with X. In addition, we investigate probabilistic extensions of the degenerate cosine-Euler (sine-Euler) polynomials, the degenerate cosine-Bernoulli (sine-Bernoulli) polynomials and the degenerate cosine (sine) polynomials. The aim of this paper is to derive several fundamental properties, identities and explicit expressions for those probabilistic extensions of special polynomials by using generating functions.

Suggested Citation

  • Wencong Liu & Taekyun Kim & Dae San Kim & Yuankui Ma, 2025. "Probabilistic Degenerate Bernoulli And Euler Polynomials Of Complex Variable," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 33(08), pages 1-11.
    • Handle: RePEc:wsi:fracta:v:33:y:2025:i:08:n:s0218348x2540167x
    DOI: 10.1142/S0218348X2540167X
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