Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)
MÁTYÁS F., Two problems related to the Bernoulli numbers
Two problems related to the Bernoulli numbers FERENC MÁTYÁS* Abstract. In this paper we deal with two similar problems. First we look for those polynomials /jt(n) with rational coefficients for which the equality S k (n)—í k+2 k H \-n k = (fk(n)) m holds for every positive integer n with some positive integer k and m(> 2). In our first theorem we prove for m> 2 that S k (n)=( fk(n)) m holds for every positive integer n if and only if m— 2, k— 3 and J 3(n)— fn 2 + \n. In the second part of this paper we look for those polynomials f(n) with complex coefficients for which the equality 2n — 2 P k(n,c) = £ n^( 2"1)B 2 n_,=(/(n)r j=k holds for every integer n>k with some integer m> 2, where /c€{2,3,4}, Bj is the j th Bernoulli number and c is a complex parameter. In our second theorem we prove for m> 2 that P 2(n,c)=(/(ri)) m holds for every integer n> 2 if and only if m = 2, c—l±i2\/2 and /(n)=n+p where p— — while in the cases of 3 or 4 and m> 2 the equality P k(n,c)—{ f(n)) m doesn't hold for any polynomial /(n). Let us introduce the following notations: (£) is the usual binomial coefficient; Bj is the j t h Bernoulli number defined by the recursion (1) EÍJ^O (k > 2) 3=0 with B 0 = 1. S k(n) = l k + 2 k + • • • + n k (n > 1, k > 1 are integers); 2n — 2 P k(n,c) = Yh 7 1 2n— j ( 2 UJ 1) - j > where n> k >2 are integers and c is a j=k complex parameter; and /(n) are polynomials of n with rational and complex coefficients, respectively. The problem of looking for those polynomials f k(n ) and integers m > 2 for which S k{n ) = f° r every positive integer n was proposed and Research supported by the Hungarian OTKA foundation, N £ T 020295.
