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A generalization of power and alternating power sums to any Appell polynomials

Kim Dae San (Sogang University, Department of Mathematics, Seoul, Republic of Korea)
Kim Taekyun (Kwangwoon University, Department of Mathematics, Seoul, Republic of Korea)

The classical power sum and alternating power sum identities can be stated as Σm,i=0 sn(i)= 1/n+1 (Bn+1(m+1)- Bn+1), Σm,i=0(-1)i sn (i) = 1/2 ((-1)m En(m+1) + En), where sn(x)=xn is the simplest possible Appell polynomial for the Sheffer pair (1,t). The impetus for this research starts from the question that what if we replace sn(x)=xn by any Appell polynomial. In this paper, we give a generalization of power and alternating power sums to any Appell polynomials.

Keywords: power sum, alternating power sum, Appell polynomial, Barnes’ multiple Bernoulli and Appell mixed-type polynomial, Barnes’ multiple Euler and Appell mixed-type polynomial