A direct method for evaluating some nice Hankel determinants and proofs of several conjectures
Let \(C_n=\frac{1}{n+1}C_{2n}^n\) be Catalan numbers, where \(C_{2n}^n\) is a binomial coefficient. Let \(M_n\) denote the Motzkin numbers genrated by the relations \(M_0=1\), \(M_{n-1}+\sum_{i=0}^{n-2} M_iM_{n-2-i}\). Let \(S_{n}\) denote the Schroeder numbers defined by the relations \(S_0=1\), \(S_n=S_{n-1}+\sum_{i=0}^{n-1}S_iS_{n-1-i}\). The following formulas for Hankel determinants are known \[ \begin{aligned}\det(C_{i+j})_{i,j=1}^n=1,\quad \det(C_{i+j+1})_{i,j=1}^{n-1}&=1,\quad \det(C_{i+j+2})_{i,j=1}^{n-1}=n+1,\\ \det(M_{i+j})_{i,j=1}^n=1,\quad \det(M_{i+j+1})_{i,j=1}^{n-1}&=\mathrm{Fib}_{n+1}(1,-1),\quad \det(M_{i+j+2})_{i,j=1}^{n-1}=\sum_{j=0}^n(\mathrm{Fib}_{j+1}(1,-1))^2,\end{aligned} \] where \(\mathrm{Fib}_n(x,s)\) is a Fibonacci polynomial defined by the relation \(\mathrm{Fib}_{n}(x,s)=x\mathrm{Fib}_{n-1}(x,s)+sFib_{n-2}(x,s)\) and \(\mathrm{Fib}_0(x,s)=1\). \[ \det(S_{i+j})_{i,j=1}^n=2^{C_{n}^2},\quad \det(S_{i+j+1})_{i,j=1}^{n-1}=2^{C_{n+1}^2},\quad \det(S_{i+j+2})_{i,j=1}^{n 1}=(2^{n+1}-1)2^{c_{2n}^n}. \] In the present paper the following sequences of numbers \(c(n,m,a,b)\), \(C(n,m,a,b,t)\), \(g(n,m,a,b)\) are considered, whose generating functions are \[ f_m(z,a,b)=\sum_{n\geq 0}c(n,m,a,b)z^n=1+azf_m(z,a,b)+bz^mf(z,a,b)^2, \] \[ \begin{multlined} F_m(z,a,b,t)=\sum_{n\geq 0}C(n,m,a,b,t)z^n=1+(a+b)z\\ F_m(z,a,b,t)+bz^mF_m(z,a,b,t)f_m(z,a,b),\end{multlined} \] \[ F_m(z,a,b)=\sum_{n\geq 0} g(n,m,a,b)z^n=1+azG_m(z,a,b)+2bz^mG_m(z,a,b)f_m(z,a,b). \] These numbers can be interpreted as weights of some lattice path sets. The sequences \(c(n,1,0,1)\), \(c(n,2,1,1)\), \(c(n,1,1,1)\) give Catalan, Motzkin and Schröder numbers. Consider determinants \[ d_r^{m}(n,a,b)=\det(c(i+j+r,m,a,b))_{i,j=0}^{n-1}, \] \[ D_r^m(n,a,b,t)=\det(C(i+t+r,m,a,b,t))_{i,j=0}^{n-1}, \] \[ dd_r^m=\det(g(i+j+r,m,a,b))_{i,j=0}^{n-1}. \] \textit{J. Cigler} [``Some nice Hankel determinants'', Preprint, \url{arXiv:1109.1449}] calculated these determinans for \(r=0,1,2\) using in some particular cases using the technique of orthogonal poynomials. In the present paper these determinants are computed using a direct method in general case.
scientific article; zbMATH DE number 6138433
scientific article