Several expressions of truncated Bernoulli-Carlitz and truncated Cauchy-Carlitz numbers

Takao Komatsu; Wenpeng Zhang; Takao Komatsu; Wenpeng Zhang

doi:10.3934/math.2020380

AIMS Mathematics

2020, Volume 5, Issue 6: 5939-5954. doi: 10.3934/math.2020380

Previous Article Next Article

Research article

Several expressions of truncated Bernoulli-Carlitz and truncated Cauchy-Carlitz numbers

Takao Komatsu ^{1
,
,},
Wenpeng Zhang ²

1.
Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou, 310018, China
2.
School of Mathematics, Northwest University, Xi’an, 710127, China

Received: 10 May 2020 Accepted: 09 July 2020 Published: 17 July 2020
MSC : 05A15, 05A19, 11A55, 11B68, 11B75, 11C20, 11R58, 11T55, 15A15

The truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers are defined as analogues of hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers, and as extensions of Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of incomplete Stirling-Carlitz numbers. In this paper, we give several expressions of truncated Bernoulli-Carlitz numbers and truncated Cauchy-Carlitz numbers as natural extensions. One kind of expressions is in continued fractions. Another is in determinants originated in Glaisher, giving several interesting determinant expressions of numbers, including Bernoulli and Cauchy numbers.
Citation: Takao Komatsu, Wenpeng Zhang. Several expressions of truncated Bernoulli-Carlitz and truncated Cauchy-Carlitz numbers[J]. AIMS Mathematics, 2020, 5(6): 5939-5954. doi: 10.3934/math.2020380

Related Papers:

Abstract

The truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers are defined as analogues of hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers, and as extensions of Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of incomplete Stirling-Carlitz numbers. In this paper, we give several expressions of truncated Bernoulli-Carlitz numbers and truncated Cauchy-Carlitz numbers as natural extensions. One kind of expressions is in continued fractions. Another is in determinants originated in Glaisher, giving several interesting determinant expressions of numbers, including Bernoulli and Cauchy numbers.

References

[1]	" L. Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math. J., 1 (1935), 137-168.
[2]	L. Carlitz, An analogue of the von Staudt-Clausen theorem, Duke Math. J., 3 (1937), 503-517. doi: 10.1215/S0012-7094-37-00340-5
[3]	L. Carlitz, An analogue of the Staudt-Clausen theorem, Duke Math. J., 7 (1940), 62-67. doi: 10.1215/S0012-7094-40-00703-7
[4]	E. U. Gekeler, Some new identities for Bernoulli-Carlitz numbers, J. Number Theory, 33 (1989), 209-219. doi: 10.1016/0022-314X(89)90007-3
[5]	S. Jeong, M. S. Kim, J. W. Son, On explicit formulae for Bernoulli numbers and their counterparts in positive characteristic, J. Number Theory, 113 (2005), 53-68. doi: 10.1016/j.jnt.2004.08.013
[6]	J. A. Lara Rodríguez, On von Staudt for Bernoulli-Carlitz numbers, J. Number Theory, 132 (2012), 495-501. doi: 10.1016/j.jnt.2011.12.005
[7]	D. Goss, Basic Structures of Function Field Arithmetic, Springer-Verlag Berlin Heidelberg, 1998.
[8]	H. Kaneko, T. Komatsu, Cauchy-Carlitz numbers, J. Number Theory, 163 (2016), 238-254. doi: 10.1016/j.jnt.2015.11.019
[9]	A. Hassen, H. D. Nguyen, Hypergeometric Bernoulli polynomials and Appell sequences, Int. J. Number Theory, 4 (2008), 767-774. doi: 10.1142/S1793042108001754
[10]	A. Hassen, H. D. Nguyen, Hypergeometric zeta functions, Int. J. Number Theory, 6 (2010), 99-126. doi: 10.1142/S179304211000282X
[11]	F. T. Howard, Some sequences of rational numbers related to the exponential function, Duke Math. J., 34 (1967), 701-716. doi: 10.1215/S0012-7094-67-03473-4
[12]	K. Kamano, Sums of products of hypergeometric Bernoulli numbers, J. Number Theory, 130 (2010), 2259-2271. doi: 10.1016/j.jnt.2010.04.005
[13]	T. Komatsu, Hypergeometric Cauchy numbers, Int. J. Number Theory, 9 (2013), 545-560. doi: 10.1142/S1793042112501473
[14]	T. Komatsu, Truncated Bernoulli-Carlitz and truncated Cauchy-Carlitz numbers, Tokyo J. Math., 41 (2018), 541-556. doi: 10.3836/tjm/1502179245
[15]	T. Komatsu, C. Pita-Ruiz, Truncated Euler polynomials, Math. Slovaca, 68 (2018), 527-536. doi: 10.1515/ms-2017-0122
[16]	T. Arakawa, T. Ibukiyama, M. Kaneko, Bernoulli Numbers and Zeta Functions, Springer Monographs in Mathematics, 2014.
[17]	J. S. Frame, The Hankel power sum matrix inverse and the Bernoulli continued fraction, Math. Comput., 33 (1979), 815-826. doi: 10.1090/S0025-5718-1979-0521297-0
[18]	H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, New York, 1948.
[19]	M. Aoki, T. Komatsu, G. K. Panda, Several properties of hypergeometric Bernoulli numbers, J. Inequal. Appl., 2019 (2019), 1-24. doi: 10.1186/s13660-019-1955-4
[20]	J. W. L. Glaisher, Expressions for Laplace's coefficients, Bernoullian and Eulerian numbers, etc. as determinants, Messenger, 6 (1875), 49-63.
[21]	M. Aoki, T. Komatsu, Remarks on hypergeometric Cauchy numbers, Math. Rep. (Bucur.), 23 (2021), to appear.
[22]	T. Komatsu, P. Yuan, Hypergeometric Cauchy numbers and polynomials, Acta Math. Hung., 153 (2017), 382-400. doi: 10.1007/s10474-017-0744-0
[23]	T. Komatsu, V. Laohakosol, P. Tangsupphathawat, Truncated Euler-Carlitz numbers, Hokkaido Math. J., 48 (2019), 569-588. doi: 10.14492/hokmj/1573722018
[24]	T. Komatsu, H. Zhu, Hypergeometric Euler numbers, AIMS Mathematics, 5 (2020), 1284-1303. doi: 10.3934/math.2020088
[25]	T. Komatsu, Complementary Euler numbers, Period. Math. Hung., 75 (2017), 302-314. doi: 10.1007/s10998-017-0199-7
[26]	T. Komatsu, W. Zhang, Several properties of multiple hypergeometric Euler numbers, Tokyo J. Math., 42 (2019), 551-570. doi: 10.3836/tjm/1502179290
[27]	T. Komatsu, J. L. Ramirez, Some determinants involving incomplete Fubini numbers, An. Şt. Univ. Ovidius Constanţa., 26 (2018), 143-170.
[28]	T. Muir, The theory of determinants in the historical order of development, Dover Publications, 1960.
[29]	N. Trudi, Intorno ad alcune formole di sviluppo, Rendic. dell' Accad. Napoli, 1862, 135-143.
[30]	F. Brioschi, Sulle funzioni Bernoulliane ed Euleriane, Ann. Mat. Pur. Appl., 1 (1858), 260-263." doi: 10.1007/BF03197335

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)