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AIMS Mathematics

2020, Volume 5, Issue 2: 1284-1303. doi: 10.3934/math.2020088
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Research article

Hypergeometric Euler numbers

  • 1.
    Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China
  • 2.
    School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
  • Received: 04 October 2019 Accepted: 10 December 2019 Published: 19 January 2020

  • MSC : 11B68, 11B37, 11C20, 15A15, 33C20

  • In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We also introduce complementary hypergeometric Euler numbers and give some characteristic properties. There are strong reasons why these hypergeometric numbers are important. The hypergeometric numbers have one of the advantages that yield the natural extensions of determinant expressions of the numbers, though many kinds of generalizations of the Euler numbers have been considered by many authors.

    Citation: Takao Komatsu, Huilin Zhu. Hypergeometric Euler numbers[J]. AIMS Mathematics, 2020, 5(2): 1284-1303. doi: 10.3934/math.2020088

    Related Papers:

  • In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We also introduce complementary hypergeometric Euler numbers and give some characteristic properties. There are strong reasons why these hypergeometric numbers are important. The hypergeometric numbers have one of the advantages that yield the natural extensions of determinant expressions of the numbers, though many kinds of generalizations of the Euler numbers have been considered by many authors.


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    [1] T. M. Apostol, On the Lerch Zeta function, Pacific. J. Math. 1 (1951), 161-167. doi: 10.2140/pjm.1951.1.161
    [2] Y. Ohno, Y. Sasaki, On the parity of poly-Euler numbers, RIMS Kokyuroku Bessatsu, B32 (2012), 271-278.
    [3] Y. Ohno, Y. Sasaki, Periodicity on poly-Euler numbers and Vandiver type congruence for Euler numbers, RIMS Kokyuroku Bessatsu, B44 (2013), 205-211.
    [4] Y. Ohno, Y. Sasaki, On poly-Euler numbers, J. Aust. Math. Soc., 103 (2017), 126-144. doi: 10.1017/S1446788716000495
    [5] R. B. Corcino, H. Jolany, C. B. Corcino, et al., On generalized multi poly-Euler polynomials, Fibonacci Quart., 55 (2017), 41-53.
    [6] T. Komatsu, J. L. Ramírez, V. Sirvent, A (p, q)-analogue of poly-Euler polynomials and some related polynomials, Ukrainian Math. J., arXiv:1604.03787.
    [7] 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
    [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] 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] K. Kamano, Sums of products of hypergeometric Bernoulli numbers, J. Number Theory, 130 (2010), 2259-2271. doi: 10.1016/j.jnt.2010.04.005
    [12] T. Komatsu, Hypergeometric Cauchy numbers, Int. J. Number Theory, 9 (2013), 545-560. doi: 10.1142/S1793042112501473
    [13] T. Komatsu, F. Luca, C. de J. Pita Ruiz V., Generalized poly-Cauchy polynomials and their interpolating functions, Colloq. Math., 136 (2014), 13-30. doi: 10.4064/cm136-1-2
    [14] T. Komatsu, C. de J. Pita Ruiz V., Truncated Euler polynomials, Math. Slovaca, 68 (2018), 52-7-536.
    [15] T. Komatsu, Complementary Euler numbers, Period. Math. Hungar., 75 (2017), 302-314. doi: 10.1007/s10998-017-0199-7
    [16] J. W. L. Glaisher, Expressions for Laplace's coefficients, Bernoullian and Eulerian numbers etc. as determinants, Messenger, 6 (1875), 49-63.
    [17] M. Aoki, T. Komatsu, G. K. Panda, Several properties of hypergeometric Bernoulli numbers, J. Inequal. Appl. 2019, Paper No. 113, 24.
    [18] M. Aoki, T. Komatsu, Remarks on hypergeometric Cauchy numbers, Math. Rep. (Bucur.) (to appear). arXiv:1802.05455.
    [19] O. TeichmÜller, Differentialrechung bei Charakteristik p, J. Reine Angew. Math., 175 (1936), 89-99.
    [20] R. Gottfert, H. Niederreiter, Hasse-Teichmüller derivatives and products of linear recurring sequences, Finite Fields: Theory, Applications, and Algorithms (Las Vegas, NV, 1993), Contemporary Mathematics, vol. 168, American Mathematical Society, Providence, RI, 1994, 117-125.
    [21] H. Hasse, Theorie der höheren Differentiale in einem algebraischen Funktionenkörper mit Vollkommenem Konstantenkörper bei beliebiger Charakteristik, J. Reine Angew. Math., 175 (1936), 50-54.
    [22] N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, available from http://oeis.org.
    [23] S. Koumandos, H. Laurberg Pedersen, Turán type inequalities for the partial sums of the generating functions of Bernoulli and Euler numbers, Math. Nachr., 285 (2012), 2129-2156. doi: 10.1002/mana.201100299
    [24] T. Komatsu, J. L. Ramírez, Some determinants involving incomplete Fubini numbers, An. Ştiinţ. Univ. "Ovidius" Constanţa Ser. Mat. 26 (2018), 3, 143-170.
    [25] N. Trudi, Intorno ad alcune formole di sviluppo, Rendic. dell' Accad. Napoli., (1862), 135-143.
    [26] T. Muir, The theory of determinants in the historical order of development, Four volumes, Dover Publications, New York, 1960.
    [27] F. Brioschi, Sulle funzioni Bernoulliane ed Euleriane, Rivista Bibliografica, (2011), 51-117.
    [28] T. Komatsu, V. Laohakosol, P. Tangsupphathawat, Truncated Euler-Carlitz numbers, Hokkaido Math. J., 48 (2019), 569-588. doi: 10.14492/hokmj/1573722018
    [29] T. Komatsu, W. Zhang, Several properties of multiple hypergeometric Euler numbers, Tokyo J. Math. (advance publication). https://projecteuclid.org/euclid.tjm/1552013762.
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