Meixner-Type Results for Riordan Arrays and Associated Integer Sequences

We determine which (ordinary) Riordan arrays are the coefficient arrays of a family of orthogonal polynomials. In so doing, we are led to introduce a family of polynomials, which includes the Boubaker polynomials, and a scaled version of the Chebyshev poynomials, using the techniques of Riordan arrays. We classify these polynomials in terms of the Chebyshev polynomials of the first and second kinds. We also examine the Hankel transforms of sequences associated to the inverse of the polynomial coefficient arrays, including the associated moment sequences.

[1]  Renzo Sprugnoli,et al.  Riordan arrays and combinatorial sums , 1994, Discret. Math..

[2]  C. Krattenthaler.  ADVANCED DETERMINANT CALCULUS , 1999, math/9902004.

[3]  J. Layman.  The Hankel transform and some of its properties. , 2001 .

[4]  Emeric Deutsch,et al.  Production Matrices and Riordan Arrays , 2007, math/0702638.

[5]  Ana Luzón.  Iterative processes related to Riordan arrays: The reciprocation and the inversion of power series , 2010, Discret. Math..

[6]  Louis W. Shapiro,et al.  Bijections and the Riordan group , 2003, Theor. Comput. Sci..

[7]  Wen-Jin Woan,et al.  Generating Functions via Hankel and Stieltjes Matrices , 2000 .

[8]  Hankel Matrices and Lattice Paths , 2012 .

[9]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[10]  Renzo Sprugnoli,et al.  Sequence characterization of Riordan arrays , 2009, Discret. Math..

[11]  H. Wall,et al.  Analytic Theory of Continued Fractions , 2000 .

[12]  M. Amlouk,et al.  Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition , 2007 .

[13]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[15]  Renzo Sprugnoli,et al.  The Method of Coefficients , 2007, Am. Math. Mon..

[16]  W. Gautschi.  Orthogonal Polynomials: Computation and Approximation , 2004 .

[17]  Aleksandar S. Cvetković,et al.  Catalan Numbers, the Hankel Transform, and Fibonacci Numbers , 2002 .

[18]  Renzo Sprugnoli,et al.  Left-inversion of combinatorial sums , 1998, Discret. Math..

[19]  Sung-Tae Jin.  A CHARACTERIZATION OF THE RIORDAN BELL SUBGROUP BY C-SEQUENCES , 2009 .

[20]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[21]  Louis W. Shapiro,et al.  The Riordan group , 1991, Discret. Appl. Math..

[22]  Luca Ferrari,et al.  Production matrices , 2005, Adv. Appl. Math..

[23]  An Application of Sobolev Orthogonal Polynomials to the Computation of a Special Hankel Determinant , 2010 .

[24]  T. J. Rivlin.  Chebyshev polynomials : from approximation theory to algebra and number theory , 1990 .

[25]  Hana Kim,et al.  Riordan group involutions , 2008 .

[26]  J. Meixner,et al.  Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion , 1934 .

[27]  A. Luzon,et al.  Recurrence relations for polynomial sequences via Riordan matrices , 2009 .

[28]  Ahmed Driss Aiat Hadj,et al.  On the powers and the inverse of a tridiagonal matrix , 2009, Appl. Math. Comput..

[29]  Kenneth H. Rosen,et al.  Catalan Numbers , 2002 .

[30]  Renzo Sprugnoli,et al.  On Some Alternative Characterizations of Riordan Arrays , 1997, Canadian Journal of Mathematics.

[31]  M. Anshelevich,et al.  Introduction to orthogonal polynomials , 2003 .

[32]  Paul Peart,et al.  TRIPLE FACTORIZATION OF SOME RIORDAN MATRICES , 1993 .