On hypergeometric Cauchy numbers of higher grade
In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy, and Euler numbers. Cauchy numbers can be generalized to the hypergeometric Cauchy numbers. Recently, Barman et al. study more general numbers in terms of determinants, which involve Bernoulli,...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2021-04-01
|
| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | http://www.aimspress.com/article/doi/10.3934/math.2021390?viewType=HTML |
| _version_ | 1818726954382131200 |
|---|---|
| author | Takao Komatsu Ram Krishna Pandey |
| author_facet | Takao Komatsu Ram Krishna Pandey |
| author_sort | Takao Komatsu |
| collection | DOAJ |
| description | In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy, and Euler numbers. Cauchy numbers can be generalized to the hypergeometric Cauchy numbers. Recently, Barman et al. study more general numbers in terms of determinants, which involve Bernoulli, Euler and Lehmer's generalized Euler numbers. However, Cauchy numbers and their generalizations are not involved in these generalized numbers. In this paper, we study more general numbers in terms of determinants, which involve Cauchy numbers. The motivations and backgrounds of the definition are in an operator related to graph theory. We also give several expressions and identities by Trudi's and inversion formulae. |
| first_indexed | 2024-12-17T22:06:24Z |
| format | Article |
| id | doaj.art-125de655950c4d548a3a4c963a6231f1 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-17T22:06:24Z |
| publishDate | 2021-04-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-125de655950c4d548a3a4c963a6231f12022-12-21T21:30:51ZengAIMS PressAIMS Mathematics2473-69882021-04-01676630664610.3934/math.2021390On hypergeometric Cauchy numbers of higher gradeTakao Komatsu0Ram Krishna Pandey11. Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou, 310018, China2. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee–247667, Uttarakhand, IndiaIn 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy, and Euler numbers. Cauchy numbers can be generalized to the hypergeometric Cauchy numbers. Recently, Barman et al. study more general numbers in terms of determinants, which involve Bernoulli, Euler and Lehmer's generalized Euler numbers. However, Cauchy numbers and their generalizations are not involved in these generalized numbers. In this paper, we study more general numbers in terms of determinants, which involve Cauchy numbers. The motivations and backgrounds of the definition are in an operator related to graph theory. We also give several expressions and identities by Trudi's and inversion formulae.http://www.aimspress.com/article/doi/10.3934/math.2021390?viewType=HTMLcauchy numberhypergeometric cauchy numberdeterminantrecurrence relationhypergeometric function |
| spellingShingle | Takao Komatsu Ram Krishna Pandey On hypergeometric Cauchy numbers of higher grade AIMS Mathematics cauchy number hypergeometric cauchy number determinant recurrence relation hypergeometric function |
| title | On hypergeometric Cauchy numbers of higher grade |
| title_full | On hypergeometric Cauchy numbers of higher grade |
| title_fullStr | On hypergeometric Cauchy numbers of higher grade |
| title_full_unstemmed | On hypergeometric Cauchy numbers of higher grade |
| title_short | On hypergeometric Cauchy numbers of higher grade |
| title_sort | on hypergeometric cauchy numbers of higher grade |
| topic | cauchy number hypergeometric cauchy number determinant recurrence relation hypergeometric function |
| url | http://www.aimspress.com/article/doi/10.3934/math.2021390?viewType=HTML |
| work_keys_str_mv | AT takaokomatsu onhypergeometriccauchynumbersofhighergrade AT ramkrishnapandey onhypergeometriccauchynumbersofhighergrade |