is the q-Pochhammer symbol. The function is arbitrary but assumed to have an expansion
Any such gives a sequence of q-difference polynomials.
ReferencesEdit
A. Sharma and A. M. Chak, "The basic analogue of a class of polynomials", Riv. Mat. Univ. Parma, 5 (1954) 325–337.
Ralph P. Boas, Jr. and R. Creighton Buck, Polynomial Expansions of Analytic Functions (Second Printing Corrected), (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263. (Provides a very brief discussion of convergence.)
August 25, 2023
difference, polynomial, combinatorial, mathematics, harmonic, polynomials, polynomial, sequence, defined, terms, derivative, they, generalized, type, brenke, polynomial, generalize, appell, polynomials, also, sheffer, sequence, definition, editthe, satisfy, re. In combinatorial mathematics the q difference polynomials or q harmonic polynomials are a polynomial sequence defined in terms of the q derivative They are a generalized type of Brenke polynomial and generalize the Appell polynomials See also Sheffer sequence Definition EditThe q difference polynomials satisfy the relation d d z q p n z p n q z p n z q z z q n 1 q 1 p n 1 z n q p n 1 z displaystyle left frac d dz right q p n z frac p n qz p n z qz z frac q n 1 q 1 p n 1 z n q p n 1 z where the derivative symbol on the left is the q derivative In the limit of q 1 displaystyle q to 1 this becomes the definition of the Appell polynomials d d z p n z n p n 1 z displaystyle frac d dz p n z np n 1 z Generating function EditThe generalized generating function for these polynomials is of the type of generating function for Brenke polynomials namely A w e q z w n 0 p n z n q w n displaystyle A w e q zw sum n 0 infty frac p n z n q w n where e q t displaystyle e q t is the q exponential e q t n 0 t n n q n 0 t n 1 q n q q n displaystyle e q t sum n 0 infty frac t n n q sum n 0 infty frac t n 1 q n q q n Here n q displaystyle n q is the q factorial and q q n 1 q n 1 q n 1 1 q displaystyle q q n 1 q n 1 q n 1 cdots 1 q is the q Pochhammer symbol The function A w displaystyle A w is arbitrary but assumed to have an expansion A w n 0 a n w n with a 0 0 displaystyle A w sum n 0 infty a n w n mbox with a 0 neq 0 Any such A w displaystyle A w gives a sequence of q difference polynomials References EditA Sharma and A M Chak The basic analogue of a class of polynomials Riv Mat Univ Parma 5 1954 325 337 Ralph P Boas Jr and R Creighton Buck Polynomial Expansions of Analytic Functions Second Printing Corrected 1964 Academic Press Inc Publishers New York Springer Verlag Berlin Library of Congress Card Number 63 23263 Provides a very brief discussion of convergence Retrieved from https en wikipedia org w index php title Q difference polynomial amp oldid 1045519217, wikipedia, wiki, book, books, library,