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Reciprocal formulae for convolutions of Bernoulli and Euler polynomials. (English) Zbl 1270.11017
The author computes the bivariate exponential generating functions for the \(\Omega\)-functions and explores their applications to the Miki-like convolution identities on Bernoulli and Euler polynomials, being able to find four classes of reciprocal summation formulae whose convolutions not only recover some well-known results but also give rise to numerous new identities concerning both Bernoulli and Euler polynomials.
The paper is organized as follows.
The Introduction describes the progress in the identities on binomial and ordinary convolutions of Bernoulli numbers recalling a vast mathematical literature consisting, e.g., of K.-W. Chen [Pac. J. Math. 208, No. 1, 39–52 (2003; Zbl 1057.11012)], of K. Shiratani and S. Yokoyama [Mem. Fac. Sci., Kyushu Univ., Ser. A 36, 73–83 (1982; Zbl 0507.12011)], of H. Gopalkrishna Gadiyar and R. Padma [“A comment on Matiyasevich’s identity \(\#0102\) with Bernoulli numbers”, arXiv:math/0608675] and of M. C. Crabb [Glasg. Math. J. 47, No. 2, 327–328 (2005; Zbl 1160.11312)]. The author remarks that the motivation for defining the bivariate \(\Omega_{m,n} (x, y)\)-sequence and the dual \(\Omega^*_{m,n} (x, y)\)-sequence is a work by H. Pan and Z. Sun [J. Comb. Theory, Ser. A 113, No. 1, 156–175 (2006; Zbl 1085.05017)]. The generating functions are recalled as a traditional method to treat combinatorial identities (see, e.g., H. S. Wilf [Generatingfunctionology. 2nd ed. Boston, MA: Academic Press (1994; Zbl 0831.05001)]) and presented as a natural choice for convolutions of Bernoulli and Euler polynomials by the author who already used them in other two related works (the author and C. Wang [Result. Math. 55, No. 1–2, 65–77 (2009; Zbl 1198.11021); Integral Transforms Spec. Funct. 21, No. 5–6, 437–457 (2010; Zbl 1214.11030)]).
The Section 1 defines both \(\Omega_{m,n} (x, y)\) and \(\Omega^*_{m,n} (x, y)\) sequences through binomial sums. Then their symmetric differences are expressed through bivariate exponential generating functions. Thanks to a relation established by the author and P. Magli [Eur. J. Comb. 28, No. 3, 921–930 (2007; Zbl 1125.05012)], which extends and unifies some results related to Bernoulli and Euler polynomials due to Z. Sun [Fibonacci Q. 39, No. 4, 324–333 (2001; Zbl 0987.05013)], to K.-J. Wu, Z.-W. Sun and H. Pan [Fibonacci Q. 42, No. 4, 295–299 (2004; Zbl 1064.11019)] and to Z.-W. Sun [Eur. J. Comb. 24, No. 6, 709–718 (2003; Zbl 1024.05010)], the author finds a new theorem fundamental in the rest of the paper.
Section 2 reviews some basic properties of Bernoulli and Euler polynomials, suggesting a more comprehensive coverage available in [M. Abramowitz (ed.) and I. A. Stegun (ed.), Handbook of mathematical functions with formulas, graphs, and mathematical tables. 10th printing. New York etc.: John Wiley & Sons (1972; Zbl 0543.33001)], in [K. H. Rosen (ed.) et al., Handbook of discrete and combinatorial mathematics. Boca Raton, FL: CRC Press (2000; Zbl 1044.00002)] and in [L. Comtet, Advanced combinatorics. The art of finite and infinite expansions. Dordrecht etc.: D. Reidel (1974; Zbl 0283.05001)].
Section 3 introduces two binomial sums whose reduction formulae are derived as preparation for computing convolutions of Bernoulli and Euler polynomials. In the proof the author employs the Chu-Vandermonde convolution formula, a partial fraction decomposition and an identity on binomial coefficients from [R. L. Graham et al., Concrete mathematics: a foundation for computer science. 2nd ed. Amsterdam: Addison-Wesley (1994; Zbl 0836.00001)].
The main body of the paper ranges from Section 4 to Section 7 where several reciprocal summation formulae as well as Miki-like identities for Bernoulli and Euler polynomials are established. The author recovers not only the so-called Miki’s identity found by H. Miki [J. Number Theory 10, 297–302 (1978; Zbl 0379.10007)] but also: a convolution identity supplied by K. Dilcher [J. Number Theory 60, No. 1, 23–41 (1996; Zbl 0863.11011)], a formula provided by G. V. Dunne and C. Schubert [“Bernoulli number identities from quantum field theory”, arXiv:math/0406610, see Commun. Number Theory Phys. 7, No. 2, 225–249 (2013; Zbl 1297.11009)], the identity \(\#0202\) from Yu. Matiyasevich [“Identities with Bernoulli numbers”, http://logic.pdmi.ras.ru/~yumat/Journal/Bernoulli/bernulli.htm], an identity from C. Faber and R. Pandharipande [Invent. Math. 139, No. 1, 173–199 (2000; Zbl 0960.14031)] and another identity due to Euler and Ramanujan and studied by I. M. Gessel [J. Number Theory 110, No. 1, 75–82 (2005; Zbl 1073.11013)]. The author points out the implicit necessity of a symmetric relation on convolutions of Bernoulli numbers due to C. F. Woodcock [J. Lond. Math. Soc., II. Ser. 20, 101–108 (1979; Zbl 0406.12009)]. In a proof passage an identity on Genocchi numbers, which resembles Euler’s identity on Bernoulli numbers, is established too.
Section 8 concludes the paper by deriving further summation formulae on Bernoulli and Euler polynomials through different weight factors on binomial convolutions. The author recovers two convolution identities illustrated by E. R. Hansen [A table of series and products. Englewood Cliffs, N.J. etc.: Prentice-Hall, Inc. (1975; Zbl 0438.00001)] and, as special case of three new identities, an equation previously obtained by Z.-W. Sun and H. Pan [Acta Arith. 125, No. 1, 21–39 (2006; Zbl 1153.11012)]. The author clarifies that, due to space limitation, this section sketches briefly the resulting identities without reproducing detailed proofs. The paper ends with two convolution sums whose closed forms seem hard to be found.

11B68 Bernoulli and Euler numbers and polynomials
05A19 Combinatorial identities, bijective combinatorics
05A15 Exact enumeration problems, generating functions