FOURIER SERIES OF r-DERANGEMENT AND HIGHER-ORDER DERANGEMENT FUNCTIONS
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A derangement is a permutation that has no fixed points and the derangement number dm is the number of fixed point-free permutations on an m element set. In this paper, we consider two Appell polynomials, called r-derangement and higher-order derangement polynomials which are natural companions for the two generalizations of the derangement num- bers, namely r-derangement and higher-order derangement numbers. Then we extend the restrictions of those polynomials on the unit interval by periodicity of the period 1 to the whole real line which are called the r- derangement and higher-order derangement functions. The aim of this paper is to derive Fourier series expansions of those functions and express them in term of Bernoulli functions. As immediate corollaries, we will be able to express the two Appell polynomials as linear combinations of Bernoulli polynomials.
Abstract
A derangement is a permutation that has no fixed points and the derangement number dm is the number of fixed point-free permutations on an m element set. In this paper, we consider two Appell polynomials, called r-derangement and higher-order derangement polynomials which are natural companions for the two generalizations of the derangement num- bers, namely r-derangement and higher-order derangement numbers. Then we extend the restrictions of those polynomials on the unit interval by periodicity of the period 1 to the whole real line which are called the r- derangement and higher-order derangement functions. The aim of this paper is to derive Fourier series expansions of those functions and express them in term of Bernoulli functions. As immediate corollaries, we will be able to express the two Appell polynomials as linear combinations of Bernoulli polynomials.
참고문헌 (6)
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