A Determinant of the Chudnovskys Generalizing the Elliptic Frobenius-Stickelberger-Cauchy Determinantal Identity

D.V. Chudnovsky and G.V. Chudnovsky [CH] introduced a generalization of the FrobeniusStickelberger determinantal identity involving elliptic functions that generalize the Cauchy determinant. The purpose of this note is to provide a simple essentially non-analytic proof of this evaluation. This method of proof is inspired by D. Zeilberger’s creative application in [Z1]. AMS Subject Classification: Primary 05A, 11A, 15A One of the most famous alternants is the Cauchy determinant which is only a special case of a determinant with symbolic entries: (1) det [ 1 xi − yj ] 1≤i,j≤n = (−1)n(n−1)/2 ∏ i<j(xi − xj)(yi − yj) ∏n i=1 ∏n j=1(xi − yj) . This expression lends itself to explicit formulas in Padé approximation theory and further applications in transcendental theory. On the other hand, the Cauchy determinant cannot be readily generalized to trigonometric or elliptic functions. However, its associate can. A natural elliptic generalization of the 1/x Cauchy kernel to the corresponding Riemann surface would be the Weierstraß ζ-function. Such a generalization was supplied by Frobenius and Stickelberger [FS], with references given to Euler and Jacobi. D.V. Chudnovsky and G.V. Chudnovsky [CH] introduced a generalization of the Frobenius Stickelberger determinantal identity involving elliptic functions that generalizes the Cauchy determinant. The purpose of this note is to provide a simple essentially non-analytic proof of this evaluation. This method of proof is inspired by D. Zeilberger’s creative application in [Z1]. We begin by recalling some notations. Given the Weierstraß elliptic function, ℘(z), then the Weierstraß ζ-function and σ-function are defined respectively by (2) ℘(z) = − d dz ζ(z), and ζ(z) = d dz log σ(z).