Generating matrices of Cnomial coe¢ cients and their spectra

In this paper, we consider a generalization of binomial coe¢ cients, called C– nomial coe¢ cients, dependent upon a sequence fungn, with indices in arithmetic progressions. We obtain a general recurrence relation and a generating matrix, and point out some new relationships between these coe¢ cients and the generalized Pascal matrices. Further, we obtain generating functions, combinatorial representations, and many new interesting identities and properties of these coe¢ cients. 1 Introduction Given that binomial coe¢ cients are very important tools in combinatorics, much e¤ort has been devoted to generalize these objects. We mention here the work of Bachmann [2], Carmichael [4], Fontené [7], Ward [26], as well as the more recent [8, 10, 14, 20, 21, 23, 24]. We de…ne the generalized C–nomial coe¢ cients formed with the terms of a sequence C = fCngn: for n m 1 n m C = C1C2 : : : Cn (C1C2 : : : Cn m) (C1C2 : : : Cm) with n 0 C = n n C = 1: One can de…ne the nth C-factorial by Cn! = C1C2 : : : Cn. When Cn = Fn is the Fibonacci sequence, the numbers m k F = Fn! Fm!Fn m! are called Fibonomials, and if Cn = (qn 1)=(q 1), the numbers m k q = (q m 1) (qm k+1 1) (q 1) (qk 1) are called q-binomial, or Gaussian coe¢ cients. Bachmann [2, p. 81], Carmichael [4, p. 40], and Jarden and Motzkin [14], all showed that if u = fungn is a Lucas sequence; that is, if u0 = 0; u1 = 1 and satis…es the recurrence un+2 = Aun+1 +Bun for all nonnegative integers n with some nonzero integers A and B such that the quadratic equation x2 Ax B = 0 has two distinct roots and whose