Multi-parameter families of higher-order Bernoulli polynomials and extended Stirling numbers linked to the generalized Bessel-Struve kernel

Aparna Rai; Musharraf Ali; Dojin Kim; Aparna Rai; Musharraf Ali; Dojin Kim

doi:10.3934/nhm.2026057

Networks and Heterogeneous Media

2026, Volume 21, Issue 4: 1501-1523. doi: 10.3934/nhm.2026057

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Multi-parameter families of higher-order Bernoulli polynomials and extended Stirling numbers linked to the generalized Bessel-Struve kernel

1.
Department of Mathematics, Gandhi Faiz-E-Aam College, Shahjahanpur-242001, Affiliated to Mahatma Jyotiba Phule Rohilkhand University, Bareilly 243006, India
2.
Department of Mathematics, Dongguk University, Seoul 04620, Republic of Korea

Received: 26 April 2026 Revised: 17 June 2026 Accepted: 24 June 2026 Published: 14 July 2026

In this paper, we introduce a new class of generalized Bernoulli polynomials and numbers by making use of the generalized Bessel–Struve kernel function as a generating mechanism. This combination enriches the structural and analytical characteristics of the resulting polynomial family and provides a broader extension of classical Bernoulli-type sequences. By choosing specific values of the free parameters, the classical Bernoulli polynomials appear as special cases, which confirms that the proposed framework has a unifying nature. We carry out a detailed investigation of the basic structural properties, which includes generating function representations, recurrence and differential relations, and implicit summation formulas. In addition, we extend the Stirling numbers of the second kind in connection with the new polynomial families, which adds a combinatorial dimension to the study. Explicit expressions for the higher-order generalized sequence $ X_{n, \mu}^{b, c, (\alpha)}(x) $ are provided for $ n = 0, 1, 2, 3 $. Numerical experiments include the verification of the addition formula to machine-precision accuracy across positive, negative, and complex inputs, systematic tracking of the real zeros as the kernel parameters vary, and a graphical analysis of the root locus and complex root distribution for degrees up to $ n = 20 $. We hope that the results obtained here will open new directions of research in mathematical analyses and related areas of applied mathematics and theoretical physics.
- Bernoulli polynomials,
- generalized Bernoulli polynomials,
- Bessel–Struve kernel function,
- Stirling numbers
Citation: Aparna Rai, Musharraf Ali, Dojin Kim. Multi-parameter families of higher-order Bernoulli polynomials and extended Stirling numbers linked to the generalized Bessel-Struve kernel[J]. Networks and Heterogeneous Media, 2026, 21(4): 1501-1523. doi: 10.3934/nhm.2026057

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Abstract

In this paper, we introduce a new class of generalized Bernoulli polynomials and numbers by making use of the generalized Bessel–Struve kernel function as a generating mechanism. This combination enriches the structural and analytical characteristics of the resulting polynomial family and provides a broader extension of classical Bernoulli-type sequences. By choosing specific values of the free parameters, the classical Bernoulli polynomials appear as special cases, which confirms that the proposed framework has a unifying nature. We carry out a detailed investigation of the basic structural properties, which includes generating function representations, recurrence and differential relations, and implicit summation formulas. In addition, we extend the Stirling numbers of the second kind in connection with the new polynomial families, which adds a combinatorial dimension to the study. Explicit expressions for the higher-order generalized sequence $ X_{n, \mu}^{b, c, (\alpha)}(x) $ are provided for $ n = 0, 1, 2, 3 $. Numerical experiments include the verification of the addition formula to machine-precision accuracy across positive, negative, and complex inputs, systematic tracking of the real zeros as the kernel parameters vary, and a graphical analysis of the root locus and complex root distribution for degrees up to $ n = 20 $. We hope that the results obtained here will open new directions of research in mathematical analyses and related areas of applied mathematics and theoretical physics.

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