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Journal articles on the topic 'Bernoulli number'

Author: Grafiati

Published: 8 November 2023

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1

Chen, Kwang-Wu. "Median Bernoulli Numbers and Ramanujan’s Harmonic Number Expansion." Mathematics 10, no. 12 (June 12, 2022): 2033. http://dx.doi.org/10.3390/math10122033.

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Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: Hn∼γ+logn−∑k=1∞Bkk·nk, where Bk is the Bernoulli numbers. In this paper, we rewrite Ramanujan’s harmonic number expansion into a similar form of Euler’s asymptotic expansion as n approaches infinity: Hn∼γ+c0(h)log(q+h)−∑k=1∞ck(h)k·(q+h)k, where q=n(n+1) is the nth pronic number, twice the nth triangular number, γ is the Euler–Mascheroni constant, and ck(x)=∑j=0kkjcjxk−j, with ck is the negative of the median Bernoulli numbers. Then, 2cn=∑k=0nnkBn+k, where Bn is the Bernoulli number. By using the result obtained, we present two general Ramanujan’s asymptotic expansions for the nth harmonic number. For example, Hn∼γ+12log(q+13)−1180(q+13)2∑j=0∞bj(r)(q+13)j1/r as n approaches infinity, where bj(r) can be determined.

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2

Jakimczuk, Rafael. "Sequences related to the e number and Bernoulli numbers." Gulf Journal of Mathematics 11, no. 1 (August 9, 2021): 38–42. http://dx.doi.org/10.56947/gjom.v11i1.666.

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3

Rawlings, Don. "Bernoulli Trials and Number Theory." American Mathematical Monthly 101, no. 10 (December 1994): 948. http://dx.doi.org/10.2307/2975160.

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4

Rawlings, Don. "Bernoulli Trials and Number Theory." American Mathematical Monthly 101, no. 10 (December 1994): 948–52. http://dx.doi.org/10.1080/00029890.1994.12004573.

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5

Kaneko, Masanobu. "Poly-Bernoulli numbers." Journal de Théorie des Nombres de Bordeaux 9, no. 1 (1997): 221–28. http://dx.doi.org/10.5802/jtnb.197.

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6

Gradl, Hans, and Sebastian Walcher. "Bernoulli algebras." Communications in Algebra 21, no. 10 (January 1993): 3503–20. http://dx.doi.org/10.1080/00927879308824745.

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7

Caratelli, Diego, Pierpaolo Natalini, and Paolo Emilio Ricci. "Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory." Symmetry 15, no. 10 (October 10, 2023): 1900. http://dx.doi.org/10.3390/sym15101900.

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Bernoulli and Euler numbers and polynomials are well known and find applications in various areas of mathematics, such as number theory, combinatorial mathematics, series expansions, and the theory of special functions. Using fractional exponential functions, we extend the classical Bernoulli and Euler numbers and polynomials to introduce their fractional-index-based types. This reveals a symmetry in relation to the classical numbers and polynomials. We demonstrate some examples of these generalized mathematical entities, which we derive using the computer algebra system Mathematica©.

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8

CRABB, M. C. "THE MIKI-GESSEL BERNOULLI NUMBER IDENTITY." Glasgow Mathematical Journal 47, no. 2 (July 27, 2005): 327–28. http://dx.doi.org/10.1017/s0017089505002545.

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9

Xu, Aimin. "Ramanujan’s Harmonic Number Expansion and Two Identities for Bernoulli Numbers." Results in Mathematics 72, no. 4 (September 18, 2017): 1857–64. http://dx.doi.org/10.1007/s00025-017-0748-7.

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10

Kargın, Levent. "p-Bernoulli and geometric polynomials." International Journal of Number Theory 14, no. 02 (February 8, 2018): 595–613. http://dx.doi.org/10.1142/s1793042118500665.

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We relate geometric polynomials and [Formula: see text]-Bernoulli polynomials with an integral representation, then obtain several properties of [Formula: see text]-Bernoulli polynomials. These results yield new identities for Bernoulli numbers. Moreover, we evaluate a Faulhaber-type summation in terms of [Formula: see text]-Bernoulli polynomials. Finally, we introduce poly-[Formula: see text]-Bernoulli polynomials and numbers, then study some arithmetical and number theoretical properties of them.

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11

Beck, Matthias, and Anastasia Chavez. "Bernoulli–Dedekind sums." Acta Arithmetica 149, no. 1 (2011): 65–82. http://dx.doi.org/10.4064/aa149-1-5.

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12

Hayes, Ben. "Harmonic models and Bernoullicity." Compositio Mathematica 157, no. 10 (August 19, 2021): 2160–98. http://dx.doi.org/10.1112/s0010437x21007442.

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We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left-orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left-orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition.

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13

KAMANO, KEN. "p-ADIC q-BERNOULLI NUMBERS AND THEIR DENOMINATORS." International Journal of Number Theory 04, no. 06 (December 2008): 911–25. http://dx.doi.org/10.1142/s179304210800181x.

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We define p-adic q-Bernoulli numbers by using a p-adic integral. These numbers have good properties similar to those of the classical Bernoulli numbers. In particular, they satisfy an analogue of the von Staudt–Clausen theorem, which includes information of denominators of p-adic q-Bernoulli numbers.

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14

DAI, LEI, and HAO PAN. "CLOSED FORMS FOR DEGENERATE BERNOULLI POLYNOMIALS." Bulletin of the Australian Mathematical Society 101, no. 2 (January 10, 2020): 207–17. http://dx.doi.org/10.1017/s0004972719001266.

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Qi and Chapman [‘Two closed forms for the Bernoulli polynomials’, J. Number Theory159 (2016), 89–100] gave a closed form expression for the Bernoulli polynomials as polynomials with coefficients involving Stirling numbers of the second kind. We extend the formula to the degenerate Bernoulli polynomials, replacing the Stirling numbers by degenerate Stirling numbers of the second kind.

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15

Jha, Sumit Kumar. "A new explicit formula for Bernoulli numbers involving the Euler number." Moscow Journal of Combinatorics and Number Theory 8, no. 4 (October 11, 2019): 385–87. http://dx.doi.org/10.2140/moscow.2019.8.389.

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16

Jan A. Grzesik. "Contour Integration Underlies Fundamental Bernoulli Number Recurrence." Real Analysis Exchange 41, no. 2 (2016): 351. http://dx.doi.org/10.14321/realanalexch.41.2.0351.

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17

Chen, Kwang-Wu. "A summation on Bernoulli numbers." Journal of Number Theory 111, no. 2 (April 2005): 372–91. http://dx.doi.org/10.1016/j.jnt.2004.08.011.

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18

Slavutskii, I. S. "A Note On Bernoulli Numbers." Journal of Number Theory 53, no. 2 (August 1995): 309–10. http://dx.doi.org/10.1006/jnth.1995.1093.

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19

Bayad, Abdelmejid, and Matthias Beck. "Relations for Bernoulli–Barnes numbers and Barnes zeta functions." International Journal of Number Theory 10, no. 05 (July 15, 2014): 1321–35. http://dx.doi.org/10.1142/s1793042114500298.

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The Barnes ζ-function is [Formula: see text] defined for [Formula: see text], Re (x) > 0, and Re (z) > n and continued meromorphically to ℂ. Specialized at negative integers -k, the Barnes ζ-function gives [Formula: see text] where Bk(x; a) is a Bernoulli–Barnes polynomial, which can be also defined through a generating function that has a slightly more general form than that for Bernoulli polynomials. Specializing Bk(0; a) gives the Bernoulli–Barnes numbers. We exhibit relations among Barnes ζ-functions, Bernoulli–Barnes numbers and polynomials, which generalize various identities of Agoh, Apostol, Dilcher, and Euler.

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20

Kim, Taekyun, and Seog-Hoon Rim. "A note on p-adic Carlitz's q-Bernoulli numbers." Bulletin of the Australian Mathematical Society 62, no. 2 (October 2000): 227–34. http://dx.doi.org/10.1017/s0004972700018700.

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In a recent paper I have shown that Carlitz's q-Bernoulli number can be represented as an integral by the q-analogue μq of the ordinary p-adic invariant measure. In the p-adic case, J. Satoh could not determine the generating function of q-Bernoulli numbers. In this paper, we give the generating function of q-Bernoulli numbers in the p-adic case.

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21

Bényi, Beáta, and Péter Hajnal. "Combinatorics of poly-Bernoulli numbers." Studia Scientiarum Mathematicarum Hungarica 52, no. 4 (December 2015): 537–58. http://dx.doi.org/10.1556/012.2015.52.4.1325.

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The Bn(k) poly-Bernoulli numbers — a natural generalization of classical Bernoulli numbers (Bn = Bn(1)) — were introduced by Kaneko in 1997. When the parameter k is negative then Bn(k) is a nonnegative number. Brewbaker was the first to give combinatorial interpretation of these numbers. He proved that Bn(−k) counts the so called lonesum 0–1 matrices of size n × k. Several other interpretations were pointed out. We survey these and give new ones. Our new interpretation, for example, gives a transparent, combinatorial explanation of Kaneko’s recursive formula for poly-Bernoulli numbers.

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22

Maulidi, Ikhsan, Vina Apriliani, and Muhamad Syazali. "Fungsi Zeta Riemann Genap Menggunakan Bilangan Bernoulli." Desimal: Jurnal Matematika 2, no. 1 (February 4, 2019): 43–47. http://dx.doi.org/10.24042/djm.v2i1.3589.

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In this article, we study about the value of Riemann Zeta Function for even numbers using Bernoulli number. First, we give some basic theory about Bernoulli number and Riemann Zeta function. The method that used in this research was literature study. From our analysis, we have a theorem to evaluate the value of Riemann Zeta function for the even numbers with its proving.

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23

Caira, R., and F. Dell'Accio. "Shepard--Bernoulli operators." Mathematics of Computation 76, no. 257 (January 1, 2007): 299–322. http://dx.doi.org/10.1090/s0025-5718-06-01894-1.

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24

Wang, Weiping. "Generalized higher order Bernoulli number pairs and generalized Stirling number pairs." Journal of Mathematical Analysis and Applications 364, no. 1 (April 2010): 255–74. http://dx.doi.org/10.1016/j.jmaa.2009.10.023.

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25

Ait Aoudia, Djilali, and Éric Marchand. "On the number of runs for Bernoulli arrays." Journal of Applied Probability 47, no. 2 (June 2010): 367–77. http://dx.doi.org/10.1239/jap/1276784897.

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We introduce and motivate the study of (n + 1) × r arrays X with Bernoulli entries Xk,j and independently distributed rows. We study the distribution of which denotes the number of consecutive pairs of successes (or runs of length 2) when reading the array down the columns and across the rows. With the case r = 1 having been studied by several authors, and permitting some initial inferences for the general case r > 1, we examine various distributional properties and representations of Sn for the case r = 2, and, using a more explicit analysis, the case of multinomial and identically distributed rows. Applications are also given in cases where the array X arises from a Pólya sampling scheme.

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26

Ait Aoudia, Djilali, and Éric Marchand. "On the number of runs for Bernoulli arrays." Journal of Applied Probability 47, no. 02 (June 2010): 367–77. http://dx.doi.org/10.1017/s0021900200006690.

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We introduce and motivate the study of (n + 1) × r arrays X with Bernoulli entries X k,j and independently distributed rows. We study the distribution of which denotes the number of consecutive pairs of successes (or runs of length 2) when reading the array down the columns and across the rows. With the case r = 1 having been studied by several authors, and permitting some initial inferences for the general case r > 1, we examine various distributional properties and representations of S n for the case r = 2, and, using a more explicit analysis, the case of multinomial and identically distributed rows. Applications are also given in cases where the array X arises from a Pólya sampling scheme.

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27

Kulkarni, Manisha, and B. Sury. "Diophantine equations with Bernoulli polynomials." Acta Arithmetica 116, no. 1 (2005): 25–34. http://dx.doi.org/10.4064/aa116-1-3.

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28

Shea, Stephen. "Finitarily Bernoulli factors are dense." Fundamenta Mathematicae 223, no. 1 (2013): 49–54. http://dx.doi.org/10.4064/fm223-1-3.

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29

Fu, Amy M., Hao Pan, and Iris F. Zhang. "Symmetric identities on Bernoulli polynomials." Journal of Number Theory 129, no. 11 (November 2009): 2696–701. http://dx.doi.org/10.1016/j.jnt.2009.05.018.

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30

Sun, Zhi-Hong. "Super congruences concerning Bernoulli polynomials." International Journal of Number Theory 11, no. 08 (November 5, 2015): 2393–404. http://dx.doi.org/10.1142/s1793042115501110.

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Let p > 3 be a prime, and let a be a rational p-adic integer. Let {Bn(x)} denote the Bernoulli polynomials given by B0= 1, [Formula: see text] and [Formula: see text]. In this paper, using Bernoulli polynomials we establish congruences for [Formula: see text] and [Formula: see text]. As a consequence we solve the following conjecture of Z. W. Sun: [Formula: see text] where [Formula: see text].

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31

Thangadurai, R. "Adams theorem on Bernoulli numbers revisited." Journal of Number Theory 106, no. 1 (May 2004): 169–77. http://dx.doi.org/10.1016/j.jnt.2003.12.006.

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32

Agoh, Takashi, and Karl Dilcher. "Higher-order recurrences for Bernoulli numbers." Journal of Number Theory 129, no. 8 (August 2009): 1837–47. http://dx.doi.org/10.1016/j.jnt.2009.02.015.

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33

Dilcher, Karl. "Sums of Products of Bernoulli Numbers." Journal of Number Theory 60, no. 1 (September 1996): 23–41. http://dx.doi.org/10.1006/jnth.1996.0110.

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34

Huang, I.-Chiau, and Su-Yun Huang. "Bernoulli Numbers and Polynomials via Residues." Journal of Number Theory 76, no. 2 (June 1999): 178–93. http://dx.doi.org/10.1006/jnth.1998.2364.

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35

HASSEN, ABDUL, and HIEU D. NGUYEN. "HYPERGEOMETRIC BERNOULLI POLYNOMIALS AND APPELL SEQUENCES." International Journal of Number Theory 04, no. 05 (October 2008): 767–74. http://dx.doi.org/10.1142/s1793042108001754.

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There are two analytic approaches to Bernoulli polynomials Bn(x): either by way of the generating function zexz/(ez - 1) = ∑ Bn(x)zn/n! or as an Appell sequence with zero mean. In this article, we discuss a generalization of Bernoulli polynomials defined by the generating function zN exz/(ez - TN-1(z)), where TN(z) denotes the Nth Maclaurin polynomial of ez, and establish an equivalent definition in terms of Appell sequences with zero moments in complete analogy to their classical counterpart. The zero-moment condition is further shown to generalize to Bernoulli polynomials generated by the confluent hypergeometric series.

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36

Young, Paul Thomas. "Bernoulli and poly-Bernoulli polynomial convolutions and identities of p-adic Arakawa–Kaneko zeta functions." International Journal of Number Theory 12, no. 05 (May 10, 2016): 1295–309. http://dx.doi.org/10.1142/s1793042116500792.

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We evaluate the ordinary convolution of Bernoulli polynomials in closed form in terms of poly-Bernoulli polynomials. As applications we derive identities for [Formula: see text]-adic Arakawa–Kaneko zeta functions, including a [Formula: see text]-adic analogue of Ohno’s sum formula. These [Formula: see text]-adic identities serve to illustrate the relationships between real periods and their [Formula: see text]-adic analogues.

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37

Raza, Mohsan, Mehak Tariq, Jong-Suk Ro, Fairouz Tchier, and Sarfraz Nawaz Malik. "Starlike Functions Associated with Bernoulli’s Numbers of Second Kind." Axioms 12, no. 8 (August 3, 2023): 764. http://dx.doi.org/10.3390/axioms12080764.

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The aim of this paper is to introduce a class of starlike functions that are related to Bernoulli’s numbers of the second kind. Let φBS(ξ)=ξeξ−12=∑n=0∞ξnBn2n!, where the coefficients of Bn2 are Bernoulli numbers of the second kind. Then, we introduce a subclass of starlike functions 𝟊 such that ξ𝟊′(ξ)𝟊(ξ)≺φBS(ξ). We found out the coefficient bounds, several radii problems, structural formulas, and inclusion relations. We also found sharp Hankel determinant problems of this class.

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38

Tanner, Jonathan W., and Samuel S. Wagstaff. "New congruences for the Bernoulli numbers." Mathematics of Computation 48, no. 177 (January 1, 1987): 341. http://dx.doi.org/10.1090/s0025-5718-1987-0866120-4.

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39

Sun, Zhi-Wei, and Hao Pan. "Identities concerning Bernoulli and Euler polynomials." Acta Arithmetica 125, no. 1 (2006): 21–39. http://dx.doi.org/10.4064/aa125-1-3.

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40

Dilcher, Karl. "On multiple zeros of Bernoulli polynomials." Acta Arithmetica 134, no. 2 (2008): 149–55. http://dx.doi.org/10.4064/aa134-2-6.

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41

Komatsu, Takao, and Florian Luca. "Generalized incomplete poly-Bernoulli polynomials and generalized incomplete poly-Cauchy polynomials." International Journal of Number Theory 13, no. 02 (February 7, 2017): 371–91. http://dx.doi.org/10.1142/s1793042117500221.

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By using the restricted and associated Stirling numbers of the first kind, we define the generalized restricted and associated poly-Cauchy polynomials. By using the restricted and associated Stirling numbers of the second kind, we define the generalized restricted and associated poly-Bernoulli polynomials. These polynomials are generalizations of original poly-Cauchy polynomials and original poly-Bernoulli polynomials, respectively. We also study their characteristic and combinatorial properties.

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42

Zhang, Jing, Lixia Zhang, and Caishi Wang. "Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals." Mathematics 10, no. 15 (July 27, 2022): 2635. http://dx.doi.org/10.3390/math10152635.

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Let N be the number operator in the space H of real-valued square-integrable Bernoulli functionals. In this paper, we further pursue properties of N from a probabilistic perspective. We first construct a nuclear space G, which is also a dense linear subspace of H, and then by taking its dual G*, we obtain a real Gel’fand triple G⊂H⊂G*. Using the well-known Minlos theorem, we prove that there exists a unique Gauss measure γN on G* such that its covariance operator coincides with N. We examine the properties of γN, and, among others, we show that γN can be represented as a convolution of a sequence of Borel probability measures on G*. Some other results are also obtained.

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43

Yang, Jizhen, and Yunpeng Wang. "Congruences involving generalized Catalan numbers and Bernoulli numbers." AIMS Mathematics 8, no. 10 (2023): 24331–44. http://dx.doi.org/10.3934/math.20231240.

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<abstract><p>In this paper, we establish some congruences mod $ p^3 $ involving the sums $ \sum_{k = 1}^{p-1}k^mB_{p, k}^{2l} $, where $ p > 3 $ is a prime number and $ B_{p, k} $ are generalized Catalan numbers. We also establish some congruences mod $ p^2 $ involving the sums $ \sum_{k = 1}^{p-1}k^mB_{p, k}^{2l_1}B_{p, k-d}^{2l_2} $, where $ m, l_1, l_2, d $ are positive integers and $ 1\leq d\leq p-1 $.</p></abstract>

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44

Kim, Taekyun. "q-Bernoulli Numbers Associated with q-Stirling Numbers." Advances in Difference Equations 2008 (2008): 1–11. http://dx.doi.org/10.1155/2008/743295.

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45

Lundell, Albert T. "On the denominator of generalized Bernoulli numbers." Journal of Number Theory 26, no. 1 (May 1987): 79–88. http://dx.doi.org/10.1016/0022-314x(87)90097-7.

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46

Gekeler, Ernst-Ulrich. "Some new identities for Bernoulli-Carlitz numbers." Journal of Number Theory 33, no. 2 (October 1989): 209–19. http://dx.doi.org/10.1016/0022-314x(89)90007-3.

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47

XIA, BINZHOU, and TIANXIN CAI. "BERNOULLI NUMBERS AND CONGRUENCES FOR HARMONIC SUMS." International Journal of Number Theory 06, no. 04 (June 2010): 849–55. http://dx.doi.org/10.1142/s1793042110003265.

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Zhao established the following harmonic congruence for prime p > 3: [Formula: see text] In this note, the authors improve it to the following congruence for prime p > 5: [Formula: see text] Meanwhile, they also improve a generalization of Zhao's congruence and extend Eisenstein's congruence.

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48

Adelberg, Arnold. "Congruences ofp-adic Integer Order Bernoulli Numbers." Journal of Number Theory 59, no. 2 (August 1996): 374–88. http://dx.doi.org/10.1006/jnth.1996.0103.

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49

Satoh, Junya. "Sums of Products of Twoq-Bernoulli Numbers." Journal of Number Theory 74, no. 2 (February 1999): 173–80. http://dx.doi.org/10.1006/jnth.1998.2331.

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50

Young, Paul Thomas. "Congruences for Bernoulli, Euler, and Stirling Numbers." Journal of Number Theory 78, no. 2 (October 1999): 204–27. http://dx.doi.org/10.1006/jnth.1999.2401.

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