On hypergeometric Cauchy numbers of higher grade

Takao Komatsu; Ram Krishna Pandey; Takao Komatsu; Ram Krishna Pandey

doi:10.3934/math.2021390

AIMS Mathematics

2021, Volume 6, Issue 7: 6630-6646. doi: 10.3934/math.2021390

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Research article

On hypergeometric Cauchy numbers of higher grade

Takao Komatsu ^{1
,
,},
Ram Krishna Pandey ²

1.
Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou, 310018, China
2.
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee–247667, Uttarakhand, India

Received: 26 February 2021 Accepted: 13 April 2021 Published: 19 April 2021
MSC : 11B75, 11B37, 11C20, 15A15, 33C05

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.
- Cauchy number,
- hypergeometric Cauchy number,
- determinant,
- recurrence relation,
- hypergeometric function
Citation: Takao Komatsu, Ram Krishna Pandey. On hypergeometric Cauchy numbers of higher grade[J]. AIMS Mathematics, 2021, 6(7): 6630-6646. doi: 10.3934/math.2021390

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Abstract

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.

References

[1]	A. Adelberg, S. F. Hong, W. L. Ren, Bounds of divided universal Bernoulli numbers and universal Kummer congruences, Proc. Amer. Math. Soc., 136 (2008), 61–71. doi: 10.1090/S0002-9939-07-09025-9
[2]	M. Aoki, T. Komatsu, Remarks on hypergeometric Cauchy numbers, Math. Rep. (Bucur.), 22 (2020), 363–380.
[3]	R. Barman, T. Komatsu, Lehmer's generalized Euler numbers in hypergeometric functions, J. Korean Math. Soc., 56 (2019), 485–505.
[4]	F. Brioschi, Sulle funzioni Bernoulliane ed Euleriane, Annali di Mat., i (1858), 260–263; Opere Mat., i (1858), 343–347.
[5]	P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89–102. doi: 10.1016/0012-365X(89)90081-2
[6]	J. W. L. Glaisher, Expressions for Laplace's coefficients, Bernoullian and Eulerian numbers etc. as determinants, Messenger (2), 6 (1875), 49–63.
[7]	T. Goy, Horadam sequence through recurrent determinants of tridiagonal matrices, Kragujevac J. Math., 42 (2018), 527–532. doi: 10.5937/KgJMath1804527G
[8]	S. F. Hong, J. R. Zhao, W. Zhao, The universal Kummer congruences, J. Aust. Math. Soc., 94 (2013), 106–132. doi: 10.1017/S1446788712000493
[9]	T. Komatsu, Hypergeometric Cauchy numbers, Int. J. Number Theory, 9 (2013), 545–560. doi: 10.1142/S1793042112501473
[10]	T. Komatsu, Incomplete poly-Cauchy numbers, Monatsh. Math., 180 (2016), 271–288. doi: 10.1007/s00605-015-0810-z
[11]	T. Komatsu, Complementary Euler numbers, Period. Math. Hungar., 75 (2017), 302–314. doi: 10.1007/s10998-017-0199-7
[12]	T. Komatsu, Incomplete multi-poly-Bernoulli numbers and multiple zeta values, Bull. Malays. Math. Sci. Soc., 41 (2018), 2029–2040. doi: 10.1007/s40840-016-0443-y
[13]	T. Komatsu, Leaping Cauchy numbers, Filomat, 32 (2018), 6167–6176. doi: 10.2298/FIL1818167K
[14]	T. Komatsu, K. Liptai, I. Mező, Incomplete Cauchy numbers, Acta Math. Hungar., 149 (2016), 306–323. doi: 10.1007/s10474-016-0616-z
[15]	T. Komatsu, Y. Ohno, Congruence properties of Lehmer's generalized Euler numbers, (preprint).
[16]	T. Komatsu, C. Pita-Ruiz, Shifted Cauchy numbers, Quaest. Math., 43 (2020), 213–226. doi: 10.2989/16073606.2018.1546775
[17]	T. Komatsu, J. L. Ramírez, Some determinants involving incomplete Fubini numbers, An. Ştiinţ. Univ. "Ovidius" Constanţa Ser. Mat., 26 (2018), 147–170.
[18]	T. Komatsu, P. Yuan, Hypergeometric Cauchy numbers and polynomials, Acta Math. Hungar., 153 (2017), 382–400. doi: 10.1007/s10474-017-0744-0
[19]	T. Komatsu, H. Zhu, Hypergeometric Euler numbers, AIMS Math., 5 (2020), 1284–1303. doi: 10.3934/math.2020088
[20]	D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Ann. Math. (2), 36 (1935), 637–649. doi: 10.2307/1968647
[21]	T. Muir, The theory of determinants in the historical order of development, Four volumes, New York: Dover Publications, 1960.
[22]	The on-line encyclopedia of integer sequences. Available from: http://oeis.org.
[23]	N. Trudi, Intorno ad alcune formole di sviluppo, Rendic. dell' Accad. Napoli, (1862), 135–143.
[24]	R. A. Zatorsky, I. I. Lishchynskyy, On connection between determinants and paradeterminants, Mat. Stud., 25 (2006), 97–102. (Ukrainian)

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