Degenerate Catalan-Daehee numbers and polynomials of order $ r $ arising from degenerate umbral calculus

Hye Kyung Kim; Dmitry V. Dolgy; Hye Kyung Kim; Dmitry V. Dolgy

doi:10.3934/math.2022213

AIMS Mathematics

2022, Volume 7, Issue 3: 3845-3865. doi: 10.3934/math.2022213

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Degenerate Catalan-Daehee numbers and polynomials of order $ r $ arising from degenerate umbral calculus

Hye Kyung Kim ¹,
Dmitry V. Dolgy ^{2
,
,}

1.
Department of Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Republic of Korea
2.
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received: 16 April 2021 Accepted: 18 November 2021 Published: 10 December 2021
MSC : 11B73, 11B83, 05A19

Many mathematicians have studied degenerate versions of some special polynomials and numbers that can take into account the surrounding environment or a person's psychological burden in recent years, and they've discovered some interesting results. Furthermore, one of the most important approaches for finding the combinatorial identities for the degenerate version of special numbers and polynomials is the umbral calculus. The Catalan numbers and the Daehee numbers play important role in connecting relationship between special numbers.

In this paper, we first define the degenerate Catalan-Daehee numbers and polynomials and aim to study the relation between well-known special polynomials and degenerate Catalan-Daehee polynomials of order $ r $ as one of the generalizations of the degenerate Catalan-Daehee polynomials by using the degenerate Sheffer sequences. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the degenerate Bernoulli polynomials and numbers of order $ r $, the degenerate Euler polynomials and numbers of order $ r $, the degenerate Daehee polynomials of order $ r $, the degenerate Bell polynomials, and so on.
Citation: Hye Kyung Kim, Dmitry V. Dolgy. Degenerate Catalan-Daehee numbers and polynomials of order $ r $ arising from degenerate umbral calculus[J]. AIMS Mathematics, 2022, 7(3): 3845-3865. doi: 10.3934/math.2022213

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Abstract

Many mathematicians have studied degenerate versions of some special polynomials and numbers that can take into account the surrounding environment or a person's psychological burden in recent years, and they've discovered some interesting results. Furthermore, one of the most important approaches for finding the combinatorial identities for the degenerate version of special numbers and polynomials is the umbral calculus. The Catalan numbers and the Daehee numbers play important role in connecting relationship between special numbers.

In this paper, we first define the degenerate Catalan-Daehee numbers and polynomials and aim to study the relation between well-known special polynomials and degenerate Catalan-Daehee polynomials of order $ r $ as one of the generalizations of the degenerate Catalan-Daehee polynomials by using the degenerate Sheffer sequences. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the degenerate Bernoulli polynomials and numbers of order $ r $, the degenerate Euler polynomials and numbers of order $ r $, the degenerate Daehee polynomials of order $ r $, the degenerate Bell polynomials, and so on.

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