On certain formulas of Karlin and Szegö.

Some identities due to Karlin and Szego which provide a relationship between determinants of classical orthogonal polynomials of Wronskian and Hankel type are shown to be specializations of a general algebraic identity between minors of a matrix. Resume On montre que des familles d’identites decouvertes par Karlin et Szego, qui relient des Wronskiens et des determinants de Hankel de polynomes orthogonaux classiques, resultent par specialisation d’une identite algebrique generale entre mineurs d’une matrice.

[1]  A. Danese.  Some identities and inequalities involving ultraspherical polynomials , 1959 .

[2]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[3]  B. Leclerc,et al.  Analogues symétriques des polynômes de Bessel , 1992 .

[4]  B. Leclerc,et al.  On Identities Satisfied by Minors of a Matrix , 1993 .

[5]  Samuel Karlin,et al.  On certain determinants whose elements are orthogonal polynomials , 1960 .

[6]  George E. Forsythe.  Second order determinants of Legendre polynomials , 1951 .

[7]  P. Turán,et al.  On the zeros of the polynomials of Legendre , 1950 .

[8]  L. M. M.-T..  The Theory of Determinants, Matrices and Invariants , 1929, Nature.

[9]  L. J. Rogers.  On the Representation of Certain Asymptotic Series as Convergent Continued Fractions , 1907 .

[10]  G. Gasper.  On Two Conjectures of Askey Concerning Normalized Hankel Determinants for the Classical Polynomials , 1973 .

[11]  T. Stieltjes,et al.  Sur la réduction en fraction continue d'une série procédant suivant les puissances descendantes d'une variable , 1889 .

[12]  Explicit evaluations of Turán expressions , 1955 .

[13]  H. Skovgaard.  On inequalities of the Turán type , 1954 .

[14]  S. Milne,et al.  New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan's tau function. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[15]  R. Askey.  An inequality for the classical polynomials , 1970 .

[16]  S. Rao,et al.  On Turán’s inequality for ultraspherical polynomials , 1957 .

[17]  O. Szâsz,et al.  Inequalities concerning ultraspherical polynomials and Bessel functions , 1950 .

[18]  O. Szâsz.  Identities and inequalities concerning orthogonal polynomials and Bessel functions , 1951 .

[19]  A. Lascoux.  Inversion des matrices de Hankel , 1990 .

[20]  Merrell L. Patrick.  Some Inequalities Concerning Jacobi Polynomials , 1971 .

[21]  E. Beckenbach,et al.  Recurrent determinants of Legendre and of ultraspherical polynomials , 1951 .

[22]  George Gasper,et al.  An inequality of Turán type for Jacobi polynomials , 1972 .

[23]  G. Szegö,et al.  On an inequality of P. Turán concerning Legendre polynomials , 1948 .

[24]  C. Jacobi.  De functionibus alternantibus earumque divisione per productum e differentiis elementorum conflatum. , 1841 .

[25]  G. Szegő.  Zeros of orthogonal polynomials , 1939 .