The q-Pilbert matrix

A generalized Filbert matrix is introduced, sharing properties of the Hilbert matrix and Fibonacci numbers. Explicit formulæ are derived for the LU-decomposition and their inverses, as well as the Cholesky decomposition. The approach is to use q-analysis and to leave the justi…cation of the necessary identities to the q-version of Zeilberger’s celebrated algorithm. 1. Introduction The Filbert (=F ibonacci-Hilbert) matrix Hn = hij n i;j=1 is de…ned by hij = 1 Fi+j 1 as an analogue of the Hilbert matrix where Fn is the nth Fibonacci number. It has been de…ned and studied by Richardson [6]. In [1], Kilic and Prodinger studied the generalized matrix with entries 1 Fi+j+r , where r 1 is an integer parameter. They gave its LU factorization and, using this, computed its determinant and inverse. Also the Cholesky factorization was derived. After this generalization, Prodinger [5] de…ned a new generalization of the generalized Filbert matrix by introducing 3 additional parameters. Again, explicit formulæ for the LU-decomposition, their inverses, and the Cholesky factorization were derived. In this paper we will consider a further generalization of the generalized Filbert Matrix F with entries 1 Fi+j+r , where r 1 is an integer parameter. We de…ne the matrix Q with entries hij as follows hij = 1 Fi+j+rFi+j+r+1 : : : Fi+j+r+k 1 ; where r 1 is an integer parameter and k 0 is an integer parameter. When k = 1, we get the generalized Filbert Matrix F, as studied before. In this paper we shall derive explicit formulæ for the LU-decomposition and their inverses. Similarly to the results of [1], the size of the matrix does not really matter, and we can think about an in…nite matrix Q and restrict it whenever necessary to the …rst n rows resp. columns and write Qn. All the identities we will obtain hold for general q, and 2000 Mathematics Subject Classi…cation. 11B39. Key words and phrases. Filbert matrix, Fibonacci numbers, q-analogues, LUdecomposition, Cholesky decomposition, Zeilberger’s algorithm. 1 2 EMRAH KILIC AND HELMUT PRODINGER results about Fibonacci numbers come out as corollaries for the special choice of q. The entries of the inverse matrix Q 1 n are not closed form expressions, as in our previous paper, but can only be given as a (simple) sum. We also provide the Cholesky decomposition. Our approach will be as follows. We will use the Binet form Fn = n n = n 11 q 1 q ; with q = = = , so that = i=pq. Throughout this paper we will use the following notations: the qPochhammer symbol (x; q)n = (1 x)(1 xq) : : : (1 xq ) and the Gaussian q-binomial coe¢ cients n k = (q; q)n (q; q)k(q; q)n k : Considering the de…nitions of the matrix Q and q-Pochhammer symbol, we rewrite the matrix Q = [hij] as hij = i k(k 1) 2 k(i+j+r q k(k 1) 4 + k(i+j+r 1) 2 (1 q) (q; q)i+j+r 1 (q; q)i+j+k+r 1 : We call the matrix Hn the q-Pilbert (=Pochhammer-Hilbert) matrix. Furthermore, we will use Fibonomial coe¢ cients n k = FnFn 1 : : : Fn k+1 F1 : : : Fk : The link between the two notations is n k = k(n k) n k with q = : In the sequel, we list all our results. Proofs are given in the following section, and they are all applications of the q-version of Zeilberger’s algorithm. This link between mathematics and computer proofs makes this article an appropriate choice for the present journal. We will obtain the LU-decomposition Q = L U : Theorem 1. For 1 d n we have Ln;d = i k(d q k(n d) 2 n 1 d 1 2d+ k + r 1 d+ r n+ d+ r + k 1 n+ r 1 and its Fibonacci corollary Ln;d = n 1 d 1 2d+ k + r 1 d+ r n+ d+ k + r 1 n+ r 1 : Theorem 2. For 1 d n we have Ud;n = i k 2 (3 k) q k 2 (d+n+r 3 2 + k 2 ) r d+dr+d2 (1 q) k