Some identities of degenerate higher-order Daehee polynomials based on $ \lambda $-umbral calculus

Dojin Kim; Sangbeom Park; Jongkyum Kwon; Dojin Kim; Sangbeom Park; Jongkyum Kwon

doi:10.3934/era.2023155

Electronic Research Archive

2023, Volume 31, Issue 6: 3064-3085. doi: 10.3934/era.2023155

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Some identities of degenerate higher-order Daehee polynomials based on $ \lambda $-umbral calculus

1.
Department of Mathematics, Dongguk University, Seoul 04620, Republic of Korea
2.
Department of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea
3.
Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Republic of Korea

Received: 23 February 2023 Revised: 16 March 2023 Accepted: 20 March 2023 Published: 22 March 2023

The degenerate versions of special polynomials and numbers, initiated by Carlitz, have regained the attention of some mathematicians by replacing the usual exponential function in the generating function of special polynomials with the degenerate exponential function. To study the relations between degenerate special polynomials, $ \lambda $-umbral calculus, an analogue of umbral calculus, is intensively applied to obtain related formulas for expressing one $ \lambda $-Sheffer polynomial in terms of other $ \lambda $-Sheffer polynomials. In this paper, we study the connection between degenerate higher-order Daehee polynomials and other degenerate type of special polynomials. We present explicit formulas for representations of the polynomials using $ \lambda $-umbral calculus and confirm the presented formulas between the degenerate higher-order Daehee polynomials and the degenerate Bernoulli polynomials, for example. Additionally, we investigate the pattern of the root distribution of these polynomials.
- generating function,
- $ \lambda $-umbral calculus,
- $ \lambda $-Sheffer polynomial,
- special polynomial,
- degenerate higher-order Daehee polynomial
Citation: Dojin Kim, Sangbeom Park, Jongkyum Kwon. Some identities of degenerate higher-order Daehee polynomials based on $ \lambda $-umbral calculus[J]. Electronic Research Archive, 2023, 31(6): 3064-3085. doi: 10.3934/era.2023155

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Abstract

The degenerate versions of special polynomials and numbers, initiated by Carlitz, have regained the attention of some mathematicians by replacing the usual exponential function in the generating function of special polynomials with the degenerate exponential function. To study the relations between degenerate special polynomials, $ \lambda $-umbral calculus, an analogue of umbral calculus, is intensively applied to obtain related formulas for expressing one $ \lambda $-Sheffer polynomial in terms of other $ \lambda $-Sheffer polynomials. In this paper, we study the connection between degenerate higher-order Daehee polynomials and other degenerate type of special polynomials. We present explicit formulas for representations of the polynomials using $ \lambda $-umbral calculus and confirm the presented formulas between the degenerate higher-order Daehee polynomials and the degenerate Bernoulli polynomials, for example. Additionally, we investigate the pattern of the root distribution of these polynomials.

References

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