Hypergeometric Cauchy numbers and polynomials (Q681571)

The general hypergeometric Cauchy polynomials are defined as \[ \frac{1}{(1+t)^z}\frac{1}{_2F_1(M,N;N+1;-t)}=\sum_{n=0}^\infty c_{M,N,n}(z)\frac{t^n}{n!}, \] for \(M,N\geq 1, n\geq 0\), where \(_2F_1(a,b;c;x)\) is the Gaussian hypergeometric function. The general hypergeometric Cauchy numbers are defined as \(c_{M,N,n}=c_{M,N,n}(0)\), the classical Cauchy numbers turn out to be \(c_n=c_{1,1,n}\). The paper gives an explicit expression for \(c_{M,N,n}\) as an alternating sum of summations over \(k\)-compositions of \(n\), and an even more involved result for the higher-order generalized hypergeometric Cauchy numbers \(c_{M,N,n}^{(r)}\), which are defined through \[ \Bigl({}_2F_1(M,N;N+1;-t)\Bigl)^{-r}=\sum_{n=0}^\infty c_{M,N,n}^{(r)}\frac{t^n}{n!}. \] Then the paper provides determinants that evaluate to \(c_{M,N,n}\) (\(c_{M,N,n}^{(r)})\), respectively), generalizing some known determinant evaluations. Finally, the paper derives the value of a convolution of general hypergeometric Cauchy polynomials.