For nonnegative integers and let be the th generalized harmonic number of rank . In this paper, the authors develop polynomials of degree in the complex variable generalizing the above harmonic numbers. These polynomials are given by
The harmonic polynomials can be expressed in terms of the generalized harmonic numbers as
which is analogous to the formula relating Bernoulli polynomials and Bernoulli numbers.
In the paper, the authors prove various relations between the generalized harmonic polynomials and other interesting sequences of polynomials such as generalized Stirling polynomials, Bernoulli polynomials, multiple Gamma functions, Cauchy polynomials and Nörlund polynomials. For example, Theorem 5.1 shows that
where, as usual, . The proofs make strong use of the summation property of Riordan arrays [see L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, Discrete Appl. Math. 34, No. 1–3, 229–239 (1991; Zbl 0754.05010)].