Cholesky’s factorization of a symmetric, positive definite matrix has been studied recently in the context of factorizing the generalized Pascal and the symmetric Pascal matrices obtained from the Pascal triangle. The purpose of the paper is to get several positive definite matrices with the Cholesky triangles as the extended generalized Pascal matrices via well-known hypergeometric functions of the type \(_2F_1(a,b;c;x)\). As a result, in this paper, the authors obtain a connection between the hypergeometric functions, the Legendre polynomials, and the Delannoy numbers.
Moreover, they show that each entry of \(P_n(x,y)P_n(x,y)^T\) can be represented by the hypergeometric functions where \(\left[ P_n(x,y)\right]_{ij} = x^{i-j}y^{i+j-2} \left(^{i-1}_{j-1} \right)\) is the extended generalized Pascal matrix, as defined by Z. Zhang and M. Liu [Linear Algebra Appl. 271, 169–177 (1998; Zbl 0892.15018)]. The authors finish the paper with the announcement that the implementation of the results here obtained, using the Computer Algebra Systems (CAS), is under active consideration.