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Triangle of coefficients of generalized Bernoulli polynomials associated with a Dirichlet character modulus 8.
3

%I #4 Jul 11 2012 14:45:14

%S 2,0,6,-44,0,12,0,-220,0,20,2166,0,-660,0,30,0,15162,0,-1540,0,42,

%T -196888,0,60648,0,-3080,0,56,0,-1771992,0,181944,0,-5544,0,72,

%U 28730410,0,-8859960,0,454860,0,-9240,0,90

%N Triangle of coefficients of generalized Bernoulli polynomials associated with a Dirichlet character modulus 8.

%C Let X be a periodic arithmetical function with period m. The generalized Bernoulli polynomials B_n(X,x) attached to X are defined by means of the generating function

%C (1)... t*exp(t*x)/(exp(m*t)-1) * sum {r = 0..m-1} X(r)*exp(r*t)

%C = sum {n = 0..inf} B_n(X,x)*t^n/n!.

%C For the theory and properties of these polynomials see [Cohen, Section 9.4]. In the present case, X is chosen to be the Dirichlet character modulus 8 given by

%C (2)... X(8*n+1) = X(8*n+7) = 1; X(8*n+3) = X(8*n+5) = -1; X(2*n) = 0.

%C Cf. A153641.

%D H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.

%F TABLE ENTRIES

%F (1)... T(2*n,2*k+1) = 0, T(2*n+1,2*k) = 0;

%F (2)... T(2*n,2*k) = (-1)^(n-k-1)*C(2*n,2*k)*2*(n-k)*A000464(n-k-1);

%F (3)... T(2*n+1,2*k+1) = (-1)^(n-k-1)*C(2*n+1,2*k+1)*2*(n-k)*A000464(n-k-1);

%F where C(n,k) = binomial(n,k).

%F GENERATING FUNCTION

%F The e.g.f. for these generalized Bernoulli polynomials is

%F (4)... t*exp(x*t)*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1)

%F = sum {n = 2..inf} B_n(X,x)*t^n/n! = 2*t^2/2! + 6*x*t^3/3! + (12*x^2 - 44)*t^4/4! + ....

%F In terms of the ordinary Bernoulli polynomials B_n(x)

%F (5)... B_n(X,x) = 8^(n-1)*{B_n((x+1)/8) - B_n((x+3)/8) - B_n((x+5)/8) + B_n((x+7)/8)}.

%F The B_n(X,x) are Appell polynomials of the form

%F (6)... B_n(X,x) = sum {j = 0..n} binomial(n,j)*B_j(X,0)*x*(n-j).

%F The sequence of generalized Bernoulli numbers

%F (7)... [B_n(X,0)]n>=2 = [2,0,-44,0,2166,0,...]

%F has the e.g.f.

%F (8)... t*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1),

%F which simplifies to

%F (9)... t*sinh(t)/cosh(2*t).

%F Hence

%F (10)... B_(2*n)(X,0) = (-1)^(n+1)*2*n*A000464(n-1); B_(2*n+1)(X,0) = 0.

%F The sequence {B_(2*n)(X,0)}n>=2 is A161722.

%F RELATION WITH TWISTED SUMS OF POWERS

%F The generalized Bernoulli polynomials may be used to evaluate sums of k-th powers twisted by the function X(n). For the present case the result is

%F (11)... sum{n = 0..8*N-1} X(n)*n^k = 1^k-3^k-5^k+7^k- ... +(8*N-1)^k

%F = [B_(k+1)(X,8*N) - B_(k+1)(X,0)]/(k+1)

%F For the proof, apply [Cohen, Corollary 9.4.17 with m = 8 and x = 0].

%F MISCELLANEOUS

%F (12)... Row sums [2, 6, -32, ...] = (-1)^(1+binomial(n,2))*A109572(n)

%F = (-1)^(1+binomial(n,2))*n*A000828(n-1) = (-1)^(1+binomial(n,2))*n* 2^(n-2)*A000111(n-1).

%e The triangle begins

%e n\k|........0.......1........2.......3......4.......5.......6

%e =============================================================

%e .2.|........2

%e .3.|........0.......6

%e .4.|......-44.......0.......12

%e .5.|........0....-220........0......20

%e .6.|.....2166.......0.....-660.......0......30

%e .7.|........0...15162........0...-1540.......0.....42

%e .8.|..-196888.......0....60648.......0...-3080......0......56

%e ...

%p with(gfun):

%p for n from 2 to 10 do

%p Genbernoulli(n,x) := 8^(n-1)*(bernoulli(n,(x+1)/8)-bernoulli(n,(x+3)/8)-bernoulli(n,(x+5)/8)+bernoulli(n,(x+7)/8));

%p seriestolist(series(Genbernoulli(n,x),x,10))

%p end do;

%Y Cf. A000111, A000464, A000828, A001586, A109572, A153641, A161722.

%K easy,tabl,sign

%O 2,1

%A _Peter Bala_, Jun 17 2009