On Some Sequences of Polynomials Generating the Genocchi Numbers
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Some Sequences of Polynomials Generating the Genocchi Numbers A. K. Svinin1* 1
Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, 134 Lermontova str., Irkutsk, 664033 Russia
Received October 13, 2019; revised November 12, 2019; accepted December 18, 2019
Abstract—We consider sequences of Genocchi numbers of the first and the second kind. For these numbers, an approach based on their representation using sequences of polynomials is developed. Based on this approach, for these numbers some identities generalizing the known identities are constructed. DOI: 10.3103/S1066369X20090078 Key words: Genocchi number, Gandhi polynomial.
1. INTRODUCTION The classic Genocchi numbers (G2n )n≥1 = (1, 1, 3, 17, 155, . . .) or, in other words, the Genocchi numbers of the first order are defined in the most simple way by the exponential generating function (see, for instance, [1], [2]) t2n 2t n = t + . (1) (−1) G 2n et + 1 (2n)! n≥1
These numbers are widely applied in different mathematical areas in generation of various objects. The most known application of these numbers is, similarly to the Euler numbers, enumeration of permutations of the given kind [3]. Some papers by Dumont et al. are dedicated to this direction (see, for instance, [4]–[6]), as well as the later papers by another authors. The generating function (1) implies that the numbers G2n are connected to the Bernoulli numbers by relation G2n = (−1)n+1 2(4n − 1)B2n , these numbers are defined as t t2n t = 1 − + . B 2n et − 1 2 (2n)! n≥1
The Genocchi numbers are contained in some number triangles. As a classic example one can consider Seidel triangle consisting of numbers gn,j , where j ≥ 1, 1 ≤ n ≤ (j + 1)/2, which are defined in the following way. Let g1,1 = 1 and the rest of the numbers in the triangle be defined by the formulae gq,2j−1 , gn,2j+1 = gq,2j . gn,2j = q≥n
q≤n
In Seidel triangle, the Genocchi numbers are defined as G2n = gn,2n−1 . In addition, in Seidel triangle one can find the so-called median Genocchi numbers or the Genocchi numbers of the second kind (H2n−1 )n≥1 = (1, 2, 8, 56, . . .), namely, H2n−1 = g1,2n . It is known that these numbers have a useful application in enumerating of various objects. D. Barsky [7] and D. Dumont [4] proved that number H2n+1 is divisible by 2n for any n ≥ 0, and thus, number hn = H2n+1 /2n is integer for *
E-mail: [email protected]
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ON SOME SEQUENCES OF POLYNOMIALS
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any n ≥ 0. The numbers (hn )n≥0 are called normalized median Genocchi numbers. One can find a good description of the combinatorial sense of these numbers in [8]. When speaking about the Genocchi numbers, we have to mention classic identities with these numbers. The most known identity is the implicit recurrent Seidel relation n 2
j=0
n (−1) G2n−2j = 0 ∀n ≥ 2. 2j j
It is known that the Genocchi numbers of the first and the second kind are connected by the relation H2n−1
n−1 2 j = (−1)
j=0
n G2n−2j ∀n ≥ 1. 2j + 1
One of the ways to generate the Genocchi numbers, both of th
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