## Shifted sums of the Bernoulli numbers, reciprocity, and denominators.(English)Zbl 1503.11047

Summary: We consider the numbers $$\mathcal{B}_{r, s} = (\mathbf{B} + 1)^r \mathbf{B}^s$$ (in umbral notation $$\mathbf{B}^n = \mathbf{B}_n$$ with the Bernoulli numbers) that have a well-known reciprocity relation, which is frequently found in the literature and goes back to the 19th century. In a recent paper, self-reciprocal Bernoulli polynomials, whose coefficients are related to these numbers, appeared in the context of power sums and the so-called Faulhaber polynomials. The numbers $$\mathcal{B}_{r, s}$$ can be recursively expressed by iterated sums and differences, so it is not obvious that these numbers do not vanish in general. As a main result among other properties, we show the non-vanishing of these numbers, apart from exceptional cases. We further derive an explicit product formula for their denominators, which follows from a von Staudt-Clausen type relation.

### MSC:

 11B68 Bernoulli and Euler numbers and polynomials 11B65 Binomial coefficients; factorials; $$q$$-identities
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### References:

  Agoh, T., Dilcher, K.: Reciprocity relations for Bernoulli numbers. Amer. Math. Monthly 115, 237-244 (2008) · Zbl 1204.11048  Bachmann, P.:Niedere Zahlentheorie. Part 2. Teubner, Leipzig (1910); Parts 1 and 2 reprinted in one volume. Chelsea, New York (1968)  BAdW: Bayerische Akademie der Wissenschaften.https://www.badw.de  Carlitz, L.:Recurrences for the Bernoulli and Euler numbers. J. Reine Angew. Math. 214/215, 184-191 (1964) · Zbl 0126.26204  Clausen, T.: Lehrsatz aus einer Abhandlung ¨uber die Bernoullischen Zahlen. Astr. Nachr. 17, 351-352 (1840)  Comtet, L.: Advanced Combinatorics. D. Reidel, Dordrecht (1974)  Denneberg, D., Grabisch, M.: Interaction transform of set functions over a finite set. Inform. Sci.121, 149-170 (1999) · Zbl 1020.91005  von Ettingshausen, A.:Vorlesungen ¨uber die h¨ohere Mathematik. Vol.1. Carl Gerold, Vienna (1827)  Gessel, I.M.: Applications of the classical umbral calculus. Algebra Univers.49, 397-434 (2003) · Zbl 1092.05005  Graham, R.L., Knuth, D.E., Patashnik, O.:Concrete Mathematics. 2nd ed.. AddisonWesley, Reading, MA (1994) · Zbl 0836.00001  Hermite, C.: Extrait d’une lettre ‘a M. Borchardt. J. Reine Angew. Math.81, 93-95 (1876) · JFM 07.0131.01  Kellner, B.C.: Faulhaber polynomials and reciprocal Bernoulli polynomials. Preprint, 1-36 (2021);arXiv:2105.15025  Lehmer, D.H.: Lacunary recurrence formulas for the numbers of Bernoulli and Euler. Ann. Math. (2)36, 637-649 (1935) · Zbl 0012.15103  Lucas, ´E.: Th´eorie des Nombres. Gauthier-Villars, Paris (1891)  Seidel, L.:Ueber eine einfache Entstehungsweise der Bernoulli’schen Zahlen und einiger verwandten Reihen. Bayer. Akad. Wiss. Math.-Phys. Kl. Sitzungsber.7, 157-187 (1877); BAdW:003384831  von Staudt, K.G.C.: Beweis eines Lehrsatzes die Bernoullischen Zahlen betreffend. J. Reine Angew. Math.21, 372-374 (1840)  Stern, M.A.: Zur Theorie der Bernoulli’schen Zahlen. J. Reine Angew. Math.84, 267-270 (1878) · JFM 09.0185.01  Stern, M.A.: Beitr¨age zur Theorie der Bernoulli’schen und Euler’schen Zahlen. Abh. K¨onigl. Ges. Wiss. G¨ottingen23, 1-44 (1878)  Tits, L.: Identit´es nouvelles pour le calcul des nombres de Bernoulli. Nouv. Ann. Math · JFM 49.0167.01
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