8 research outputs found
Some determinants of path generating functions
We evaluate four families of determinants of matrices, where the entries are
sums or differences of generating functions for paths consisting of up-steps,
down-steps and level steps. By specialisation, these determinant evaluations
have numerous corollaries. In particular, they cover numerous determinant
evaluations of combinatorial numbers - most notably of Catalan, ballot, and of
Motzkin numbers - that appeared previously in the literature.Comment: 35 pages, AmS-TeX; minor corrections; final version to appear in Adv.
Appl. Mat
EVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS
Abstract. Motivated by a recent work about finite sequences where the n-th term is bounded by n
Determinants containing powers of polynomial sequences
We derive identities for the determinants of matrices whose entries are
(rising) powers of (products of) polynomials that satisfy a recurrence
relation. In particular, these results cover the cases for Fibonacci
polynomials, Lucas polynomials and certain orthogonal polynomials. These
identities naturally generalize the determinant identities obtained by Alfred,
Carlitz, Prodinger, Tangboonduangjit and Thanatipanonda.Comment: 12 page
Advanced Determinant Calculus: A Complement
This is a complement to my previous article "Advanced Determinant Calculus"
(S\'eminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the
present article, I share with the reader my experience of applying the methods
described in the previous article in order to solve a particular problem from
number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12
(2003), 441-456). Moreover, I add a list of determinant evaluations which I
consider as interesting, which have been found since the appearance of the
previous article, or which I failed to mention there, including several
conjectures and open problems.Comment: AmS-LaTeX, 85 pages; Final, largely revised versio