0,5

For v>=1 the orthogonal polynomials p(n,v,x) have v integer zeros k*(k-1), k=1..v, for every n>=v.

Coefficients of p(n,v=1,x) (in the quoted Bruschi et al. paper p^{(\nu)}_n(x) of eqs. (4) and (8a),(8b)) in increasing powers of x.

The v-family p(n,v,x) consists of characteristic polynomials of the tridiagonal M x M matrix V=V(M,v) with entries V_{m,n} given by v*(v-1)-(m-1)^2 - (v-m)^2 if n=m, m=1,...,M;(m-1)^2 if n=m-1, m=2,...,M; (v-m)^2 if n=m+1, m=1..M-1 and 0 else. p(n,v,x):=det(x*I_n - V(n,v) with the n dimensional unit matrix I_n.

p(n,v=1,x) has, for every n>=1 a zero for x=0, i.e. det(V(n,1))=0 for every n>=1. This is obvious.

The column sequences give A000007, A010790, A129460, A129461 for m=0,1,2,3.

M. Bruschi, F. Calogero and R. Droghei, Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials, J. Physics A, 40(2007)3815-3829.

W. Lang, First ten rows.

a(n,m)=[x^m]p(n,1,x), n>=0, with the three term recurrence for orthogonal polynomial systems of the form p(n,v,x) = (x+2*(n-1)^2-2*(v-1)*(n-1)-v+1)*p(n-1,v,x) -(n-1)^2*(n-1-v)^2*p(n-2,v,x), n>=1; p(-1,v,x)=0 and p(0,v,x)=1. Put v=1 here.

Recurrence: a(n,m) = a(n-1,m-1)+(2*(n-1)^2-2*(v-1)*(n-1)-v+1)*a(n-1,m) -((n-1)^2*(n-1-v)^2)*a(n-2, m); a(n,m)=0 if n<m, a(-1,m):=0, a(0,0)=1, a(n,-1)=0. Put v=1 here.

[1]; [0,1]; [0,2,1]; [0,12,10,1]; [0,144,156,28,1]; [0,2880,3696,908,60,1]; ...

n=5,[0,2880,3696,908,60,1] stands for the polynomial x*(2880+3696*x+908*x^2+60*x^3+1*x^4) with one zero 0 and some other four zeros.

Tridiagonal matrix V(5,1)=[[0,0,0,0,0],[1,-2,1,0,0],[0,4,-8,4,0],[0,0,9,-18,9],[0,0,0,16,-32]].

Row sums give A129458. Cf. A129462 (v=2 triangle).

Adjacent sequences: A129062 A129063 A129064 this_sequence A129066 A129067 A129068

Sequence in context: A119830 A039910 A129467 this_sequence A024026 A009829 A051652

nonn,tabl,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) May 04 2007