ASYMMETRIC GENERALIZATIONS OF THE FILBERT MATRIX AND VARIANTS. Emrah Kılıç and Helmut Prodinger

PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 95 (09) (04), 67 80 DOI: 098/PIM40967K ASYMMETRIC GENERALIZATIONS OF THE FILBERT MATRIX AND VARIANTS Emrah Kılıç and Helmut Prodinger Communicated by Gradimir Milovanović Abstract Four generalizations of the Filbert matrix are considered, with additional asymmetric parameter settings Explicit formulæ are derived for the LU-decompositions, their inverses, and the inverse matrix The approach is mainly to use the q-analysis and to leave the justification of the necessary identities to the q-version of Zeilberger s algorithm for some of them, and for the rest of the necessary identities, to guess the relevant quantities and proving them later by induction Introduction The Filbert matrix F N = (ȟij) N i,j= is defined by ȟij = F i+j as an analogue of the Hilbert matrix where F n is the nth Fibonacci number It has been defined and studied by Richardson [0] The Filbert matrix has been extended by Berg [] and Ismail [] The present authors have generalized and extended this concept in a series of papers [3, 4, 5, 6, 7, 9] to matrices with entries F λ(i+j)+r F λ(i+j+)+r F λ(i+j+k )+r and L λ(i+j)+r L λ(i+j+k )+r Here, λ, k and r are integer parameters and L n are Lucas numbers In another direction [5], the matrices with entries g ij = F λ(i+j)+r F λ(i+j)+s and v ij = L λ(i+j)+r L λ(i+j)+s were introduced; here s, r and λ are integer parameters such that s r, and r, s and λ This was the first nontrivial instance where the numerator of the entries is not equal to one 00 Mathematics Subject Classification: B39, 05A30, 5A3 The second author was supported by an incentive grant of the NRF South Africa 67

68 KILIÇ AND PRODINGER All these extensions were driven by the search for nice explicit results: explicit formulæ for the LU-decomposition, their inverses, and the Cholesky factorization Here, we go one step further, by allowing an asymmetric growth of indices We, however, confine ourselves to k = ; for this instance, the inverse matrix also enjoys nice closed form entries, which is no longer true for k To be more specific, we introduce four generalizations of the Filbert matrix F, and define the matrices T, M, H and W with entries t ij = F λi+µj+r, m ij = F λi+µj+r F λi+µj+s, h ij = L λi+µj+r and w ij = L λi+µj+r L λi+µj+s, respectively, where s, r, λ and µ are integer parameters such that s r, and r, s and λ, µ Of course, because of these asymmetric entries, we cannot get a Cholesky decomposition anymore Now we discuss our settings Let {U n } and {V n } be generalized Fibonacci and Lucas sequences, respectively, whose Binet forms are U n = αn β n α β qn = αn q and V n = α n + β n = α n ( + q n ) with q = β/α = α, so that α = i/ q ( ( ) ( )) When α = + 5 or equivalently q = 5 / + 5, the sequence {Un } is reduced to the Fibonacci sequence {F n } and the sequence {V n } is reduced to the Lucas sequence {L n } When α = + ( or equivalently q = ( ) / ( + )), the sequence {U n } is reduced to the Pell sequence {P n } and the sequence {V n } is reduced to the Pell-Lucas sequence {Q n } We will mostly deal with the q-forms; translating the results back to the Fibonacci and Lucas world, say, is easy: We only have to systematically replace q n by q α F n n and + q n by α n L n and replace what is eventually left by its numerical values Throughout this paper we will use the notation of the q-pochhammer symbol (x; q) n = ( x)( xq) ( xq n ) We rewrite the entries of the matrices T, M, H and W in the q-form: t ij = i r λi µj q (λi+µj+r ) q q λi+µj+r, m ij = i r s q s r q λi+µj+r q λi+µj+s, h ij = i λi µj r q (λi+µj+r) + q λi+µj+r, w ij = i r s q s r + q λi+µj+r + q λi+µj+s In each of the four instances, we consider the LU-decomposition of the matrix M = LU We are able to get explicit formulæ for L, U, L, U and M

ASYMMETRIC GENERALIZATIONS OF THE FILBERT MATRIX 69 As it was noted in the above cited earlier papers, the sizes of the matrices do not really matter, and they can be thought of as infinite matrices and we may restrict them whenever necessary to the first N rows resp columns and write T N etc This is not true for the inverse matrices; here, the entries depend on N All our identities hold for general q, and the results about Fibonacci and Lucas numbers come out as corollaries for the special choice of q, as explained The important part is to find the explicit forms This was done by experiments with a computer algebra system and spotting patterns This becomes increasingly complicated when more and more new parameters are introduced, as the guessing only works for fixed choices of the parameters, and one needs to vary them as well Once one knows how the entries look like, proofs are by reducing sums to single terms For this, the q-zeilberger algorithm is a handy tool However for the matrices M and W, the present versions of the q-zeilberger algorithm do not work, and we have to simulate it by noticing that the relevant sums are Gospersummable To do this, some more guessing (with an additional parameter) is required Consequently, since all these proofs are routine and somewhat tedious, we only present two typical examples It would be a good student project to work them all out in full detail As an illustration, we always write out the Fibonacci/Lucas cases for λ, µ {, 3} and r =, s = The matrix T We obtain the LU-decomposition T = L U: Theorem For d n we have L n,d = i λ(d n) q λ(d n) (q λ ; q λ ) n (q r+µ+λd ; q µ ) d (q λ ; q λ ) n d (q λ ; q λ ) d (q r+µ+λn ; q µ ) d Fibonacci Corollary for λ = 3, µ = and r = : Corollary For d n, L n,d = ( n ( n d )( d ) F 3t F t+3d+ )( d )( d ) F 3t F 3t F t+3d+ Theorem For d n we have U d,n = i r λd µn q µ n d +rd r + d (q λ ; q λ ) d ( q) (q r+µn+λ ; q λ ) d (q r+µ+λd ; q λ ) d (q µ ; q µ ) n (q µ ; q µ ) n d Fibonacci Corollary for λ = 3, µ = and r = :

70 KILIÇ AND PRODINGER Corollary For d n ( d )( n ) ( ) (d )+d F 3t F t U d,n = ( d )( n d )( d ) F 3t+n+ F t F t+3d+ The inverses of the matrices L and U: Theorem 3 For d n we have L n,d = ( )n d i λ(d n) q λ (n d) (q λ ; q λ ) n (q r+µ+λd ; q µ ) n (q λ ; q λ ) n d (q λ ; q λ ) d (q r+µ+λn ; q µ ) n Fibonacci Corollary for λ = 3, µ = and r = : ( )(n d+ L n,d = ( n d Corollary 3 For d n ) ( n )( n ) F 3t F t+3j+ )( d )( n ) F 3t F 3t F t+3n+ Theorem 4 For d n we have U d,n = q λ n +µ d µnd rn+ r + ( ) n+d i r +λn+µd (q r+λ+µd ; q λ ) n (q r+µ+λn ; q µ ) n ( q)(q λ ; q λ ) n (q µ ; q µ ) n d (q µ ; q µ ) d Its Fibonacci Corollary for λ = 3, µ = and r = : Corollary 4 For d n U d,n = ( )(n )+d ( n )( n ) F 3t+d+ F t+3n+ ( n )( n d )( d ) F 3t F t F t As a consequence, we can compute the determinant of T N, since it is simply evaluated as U, U N,N : Theorem 5 dett N = i (µ λ)(n+ )+N( r) q N (µ+r) N+ 6 µn(n )+ λn(n+)(n+) ( q) N N d= Fibonacci Corollary for λ = 3, µ = and r = : Corollary 5 dett N = ( ) (N+ 3 )+N N F 3d+N+ d= (q λ ; q λ ) d (q µ ; q µ ) d (q r+λ+µd ; q λ ) d (q r+µ+λd ; q µ ) d d F 3t F t F 3t+N+ F t+3d+

ASYMMETRIC GENERALIZATIONS OF THE FILBERT MATRIX 7 Now we compute the inverse of the matrix T It depends on the dimension, so we compute (T N ) Theorem 6 For n, d N: (T N ) n,d = ( )n d i r +µn+λd q µ n +λ d µnn λnd Ns+ r + (q r+µn+λ ; q λ ) N (q r+λd+µ ; q µ ) N (q λ ; q λ ) N d (q λ ; q λ ) d (q µ ; q µ ) N n (q µ ; q µ ) n q r+µn+λd q Remark The inverse matrix and the other inverse matrices presented in this paper were not computed using the inverses of L and U, but rather obtained directly by our usual guessing strategy While the first alternative would mean that we would have to simplify a sum, the second approach stays within our chosen method Fibonacci Corollary for λ = 3, µ = and r = : Corollary 6 For n, d N: ( N )( N ) ( )(n+ )+(N+)(n+)+d F t+3d+ F 3t+n+ (T N ) n,d = ( N n )( n )( N d )( d ) F n+3d+ F t F t F 3t F 3t 3 The matrix M We obtain the LU-decomposition M = L U: Theorem 3 For d n we have L n,d = (q λ ; q λ ) n (q s+λd+µ ; q µ ) d q d(d+) (q λ ; q λ ) d (q λ ; q λ ) n d (q s+λn+µ ; q µ d(d+) ) d q +r+s(d ) Fibonacci Corollary for λ =, µ = 3, r = and s = : Corollary 3 For d n, ( d )( d ) F t F 3t+d L n,d = ( d )( n d )( d ) F t F t F 3t+n Theorem 3 For d n we have +λ(n d)+r+s(d ) F 5d(d+)/+n 3d+ F 5d(d+)/ d+ U d,n = i r s d(d ) q r+s +ds (q λ ; q λ ) d (q µ ; q µ ) n (q µ ; q µ ) n d (q s+λ+nµ ; q λ ) d (q s+µ+λd ; q µ ) d (d+)d q d(d ) q s+r+ds +µ(n d)+r+s(d ) Fibonacci Corollary for λ =, µ = 3, r = and s = : ( q r s )

7 KILIÇ AND PRODINGER Corollary 3 For d n U d,n = ( ) d+ ( ) (d ) ( n d The inverses of the matrices L and U: Theorem 33 For d n we have L n,d = ( )n d q λ n(n ) +λ d(d+) λnd ( d )( n F t F 3t ) )( d )( d ) F 3t F 3t+d F t+3n F 5d(d+)/+3n 4d+ F 5d(d )/ d+3 q λd+ λ+µ n + λ µ n+r+s(n ) λ+µ q λn+ n + λ µ n+r+s(n ) (q λ ; q λ ) n (q s+λd+µ ; q µ ) n (q λ ; q λ ) d (q λ ; q λ ) n d (q s+λn+µ ; q µ ) n Fibonacci Corollary for λ =, µ = 3, r = and s = : Corollary 33 For d n ( n )( n ) ( )n d F t F 3t+d L n,d = ( d )( n d )( n ) F t F t F 3t+n Theorem 34 For d n we have F 5n(n+)/ 4n d+3 F 5n(n )/ n+3 U d,n = is r ( ) n d n(n ) λ q +µ d(d+) µnd+ r+s ns ( q r s ) (qλ+µd+s ; q λ ) n (q µ+λn+s ; q µ ) n q n(n ) (q λ ; q λ ) n (q µ ; q µ ) n d (q µ ; q µ n(n+) ) d q +r+s(n ) Its Fibonacci Corollary for λ =, µ = 3, r = and s = : Corollary 34 For d n ( n )( n ) F t+3d F 3t+n U d,n = ( )(d+ )+nd ( d )( n )( n d ) F 3t F t F 3t +µ(n d)+r+s(n ) F 5n(n )/+n 3d+3 F 5n(n+)/ n+ As a consequence, we can compute the determinant of M N, since it is simply evaluated as U, U N,N : Theorem 35 detm N = i N(r s) q 6 N(N )+ 3 N(Ns+s r) ( q r s ) N N (q λ ; q λ ) d (q µ ; q µ ) d q (q λ+dµ+s ; q λ ) d (q µ+λd+s ; q µ ) d d= (d+)d +r+s(d ) q d(d ) s+r+ds

ASYMMETRIC GENERALIZATIONS OF THE FILBERT MATRIX 73 Fibonacci Corollary for λ =, µ = 3, r = and s = : Corollary 35 detm N = ( ) N+ 6 N(N ) N F 5d(d+)/ d+ d F d= 5d(d )/ d+3 ( d ( d )( d F t F 3t ) )( d ) F 3t+d F t+3d Now we compute the inverse of the matrix M It depends on the dimension, so we compute (M N ) Theorem 36 For n, d N: (M N ) n,d = ( )n d i s r q r+s +λ d(d+) +µ n(n+) Ns Ndλ Nnµ (q s+λ+µn ; q λ ) N (q s+µ+λd ; q µ ) N (q λ ; q λ ) N d (q λ ; q λ ) d (q µ ; q µ ) n (q µ ; q µ ) N n N(N+) q ( q s+µn+λd )( q r s N(N+) ) q +r+sn s Fibonacci Corollary for λ =, µ = 3, r = and s = : Corollary 36 For n, d N: λd µn+r+sn s ( ( )( n+ N )( N ) )+d+n+nn F 3t+d F t+3n (M N ) n,d = ( d )( n )( N d )( N n F t F 3t F t F 3t )F d+3n 4 The matrix H We obtain the LU-decomposition H = L U: Theorem 4 For d n we have F 5N(N+)/ d 3n N+3 F 5N(N+)/ N+ L n,d = q λ n d i λ(n d) ( qr+µ+λd ; q µ ) d (q λ ; q λ ) n ( q r+µ+λn ; q µ ) d (q λ ; q λ ) d (q λ ; q λ ) n d Fibonacci Corollary for λ = 3, µ = and r = : Corollary 4 For d n, ( d ) L t+3d+ L n,d = ( d ) L t+3n+ ( d ( n ) F 3t )( n d ) F 3t F 3t

74 KILIÇ AND PRODINGER Theorem 4 For d n we have U d,n = ( ) d i r µn λd q µ n d +rd r + d (q λ ; q λ ) d ( q r+µn+λ ; q λ ) d ( q r+µ+λd ; q µ ) d (q µ ; q µ ) n (q µ ; q µ ) n d Fibonacci Corollary for λ = 3, µ = and r = : Corollary 4 For d n ( ) (d ) 5 d U d,n = ( d )( d ) L 3t+n+ L t+3d+ The inverses of the matrices L and U: Theorem 43 For d n we have ( d )( n ) F 3t F t ( n d ) F t L n,d = ( )n d i λ(n d) q λ (n d) ( q r+µ+λd ; q µ ) n (q λ ; q λ ) n ( q r+µ+λn ; q µ ) n (q λ ; q λ ) d (q λ ; q λ ) n d Fibonacci Corollary for λ = 3, µ = and r = : Corollary 43 For d n L n,d = ( )(n+ ( n )+( d ) nd ( n )( n ) L t+3d+ F 3t )( d )( n d ) L t+3n+ F 3t F 3t Theorem 44 For d n we have d,n = ( )d i r+λn+µd q λ n U +µ d µnd rn+ r ( qr+µd+λ ; q λ ) n ( q r+λn+µ ; q µ ) n (q λ ; q λ ) n (q µ ; q µ ) d (q µ ; q µ ) n d Its Fibonacci Corollary for λ = 3, µ = and r = : Corollary 44 For d n ( n )( n ) L t+3n+ L 3t+d+ U d,n = ( )(n+ ) d ( d )( n d )( n ) 5 n F t F t F 3t As a consequence, we can compute the determinant of H N, since it is simply evaluated as U, U N,N : Theorem 45 deth N = i N(N+)+Nr+N(N+) q N + N 4 + N3 6 + N r

ASYMMETRIC GENERALIZATIONS OF THE FILBERT MATRIX 75 N d= Fibonacci Corollary for λ = 3, µ = and r = : (q λ ; q λ ) d (q µ ; q µ ) d ( q r+µd+λ ; q λ ) d ( q r+µ+λd ; q µ ) d Corollary 45 ( d )( d ) deth N = 5 N(N )/ ( ) (N+ ) N F 3t F t ( d )( d ) d= L 3t+d+ L t+3d+ Now we compute the inverse of the matrix H It depends on the dimension, so we compute (H N ) Theorem 46 For n, d N: (H N ) n,d = ( )N n d i r+µn+λd q µ n +λ d µnn λnd Nr+ r ( q r+µn+λ ; q λ ) N ( q r+λd+µ ; q µ ) N (q λ ; q λ ) N d (q λ ; q λ ) d (q µ ; q µ ) N n (q µ ; q µ ) n + q r+µn+λd Fibonacci Corollary for λ = 3, µ = and r = : Corollary 46 For n, d N: ( N )( N ) L 3t+n+ L t+3d+ (H N ) n,d = ( )(d+ )+Nd+n+d ( N d )( d )( N n )( n ) 5 N F 3t F 3t F t F t 5 The matrix W Now we collect our results related to the matrix W For convenience, we use the same letters L, U, but with a different meaning We obtain the LU-decomposition W = L U: Theorem 5 For d n we have L n,d = (q λ ; q λ ) n (q λ ; q λ ) d (q λ ; q λ ) n d ( q s+λd+µ ; q µ ) d ( q s+λn+µ ; q µ ) d ( )d q d(d+) ( ) d q +λ(n d)+r+s(d ) d(d+) +r+s(d ) Fibonacci Corollary for λ =, µ = 3, r = and s = : Corollary 5 For d n, ( n ) d F t L 3t+d L n,d = ( d )( n d ) d F t F t L 3t+n { L5d(d+)/+n 3d+ L 5d(d+)/ d+ F 5d(d+)/+n 3d+ F 5d(d+)/ d+ if d is even, if d is odd

76 KILIÇ AND PRODINGER Theorem 5 For d n we have U d,n = is r ( ) d d(d ) + q r+s +ds (q λ ; q λ ) d (q µ ; q µ ) n (q µ ; q µ ) n d ( q λ+nµ+s ; q λ ) d ( q µ+λd+s ; q µ ) d (d+)d ( )d q + ( ) d q +µ(n d)+r+ds s d(d ) +r+ds s Fibonacci Corollary for λ =, µ = 3, r = and s = : ( q r s ) Corollary 5 For d n ( d )( n ) F t F 3t U d,n = ( ) (d ) ( n d )( d )( d ) F 3t L 3t+d L t+3n 5 d L 5(d+)d/+3n 4d+ F 5d(d )/ d+3 if d is odd, if d is even The inverses of the matrices L and U: Theorem 53 For d n we have L 5 d F 5(d+)d/+3n 4d+ L 5d(d )/ d+3 n,d = ( )n d q λ(n d ) (q λ(n d+) ; q λ ) d ( q s+µ+λd ; q µ ) n (q λ ; q λ ) d ( q s+µ+λn ; q µ ) n + n(n+) ( )n q µn λd+r s+ns + ( ) n n(n+) q Fibonacci Corollary for λ =, µ = 3, r = and s = : µn λn+r s+ns Corollary 53 For d n ( n )( n ) ( )n d F t L 3t+d n,d = ( d )( n d )( n ) F t F t L 3t+n L Theorem 54 For d n we have U d,n = q r+s r d(d+) +µ λ n(n ) ns µnd { F5n(n+)/ d 4n+3 F 5n(n+)/ 6n+3 L 5n(n+)/ d 4n+3 L 5n(n+)/ 6n+3 if n is odd, if n is even is r ( ) d ( q s+λ+µd ; q λ ) n ( q s+µ+λn ; q µ ) n ( q r s )(q λ ; q λ ) n (q µ ; q µ ) n d (q µ ; q µ ) d + ( )n q (n(n )/)+µ(n d)+r+s(n ) ( ) n n(n+) q +r+s(n ) Its Fibonacci Corollary for λ =, µ = 3, r = and s = :

ASYMMETRIC GENERALIZATIONS OF THE FILBERT MATRIX 77 Corollary 54 For d n U d,n = ( )(d+ )+d+n nd ( n ( d )( n ) L t+3d L 3t+n )( n F 3t )( n d ) F t F 3t { L5n(n )/+n 3d+3 5 n F 5n(n+)/ n+ if n is even, F 5n(n )/+n 3d+3 5 n L 5n(n+)/ n+ if n is odd As a consequence, we can compute the determinant of W N, since it is simply evaluated as U, U N,N (we only state the Fibonacci version for λ =, µ = 3, r = and s = ): Theorem 55 detw N = ( ) 6 N(N ) N d= ( d ( d )( d F t F 3t ) )( d ) L 3t+d L t+3n 5 d L 5(d+)d/ d+ F 5d(d )/ d+3 5 d F 5(d+)d/ d+ L 5d(d )/ d+3 if d is odd, if d is even Now we compute the inverse of the matrix W It depends on the dimension, so we compute (W N ) Theorem 56 For d n N: (W N ) n,d = is r ( ) n d +N q λ d(d+) +µ n(n+) Ns Ndλ Nnµ+ r+s ( q λ+µn+s ; q λ ) N ( q µ+λd+s ; q µ ) N ( + q s+µn+λd )( q r s ) (q λ ; q λ ) N d (q λ ; q λ ) d (q µ ; q µ ) n (q µ ; q µ ) N n + N(N+) ( )N q ( ) N q λd µn+r+sn s N(N+) +r+sn s Fibonacci Corollary for λ =, µ = 3, r = and s = : Corollary 55 For d n N: ( N )( N ) ( )(n+ )+n+d Nn L 3t+d L t+3n (W N ) n,d = ( d )( n )( N d )( N n F t F 3t F t F 3t )L d+3n { L5N(N+)/ d 3n N+3 5 N F 5N(N+)/ N+ if N is even, F 5N(N+)/ d 3n N+3 5 N L 5N(N+)/ N+ if N is odd

78 KILIÇ AND PRODINGER 6 Proofs Following our introductory remarks, we present now two proofs about the matrix T In order to show that indeed T = L U, we need to show that for any m, n: L m,d U d,n = t m,n = i r λm µn q (λm+µn+r ) q q λm+µn+r d In a rewritten form the formula to be proved reads L m,d U d,n = (q λ ; q λ ) m (q µ ; q µ ) n ( q)i r µn λm q (λm+µn r ) d q d +rd µd λ d (q r+µ+λd ; q µ ) d (q λ ; q λ ) m d (q r+µ+λm ; q µ ) d (q µ ; q µ ) n d (q r+λ+µn ; q λ ) d (q r+µ+λd ; q µ ) d d Apart form the constant factor, it should be proven that (6) d q d +rd µd λ d (q r+µ+λd ; q µ ) d (q λ ; q λ ) m d (q r+µ+λm ; q µ ) d (q µ ; q µ ) n d (q λ+µn+r ; q λ ) d (q r+µ+λd ; q µ ) d = (q λ ; q λ ) m (q µ ; q µ ) n ( q λm+µn+r ) For the verification of the last equation, let us denote the LHS of the equation (6) by SUM m, then the Mathematica version of the q-zeilberger algorithm [8] produces the recursion q m(λ )+µn+r SUM m = ( q m(λ ) )( q mλ+µn+r ) SUM m Now we move to the inverse matrices Since L and L are lower triangular matrices, we only need to look at the entries indexed by (m, n) with m n: n d m L m,d L d,n = (qλ ; q λ ) m (q λ ; q λ ) n i λm q λm ( ) n i λn q λn n d m ( ) d q λdn+ λd λd (q µ+λd+r ; q µ ) d (q µ+λn+r ; q µ ) d (q λ ; q λ ) m d (q µ+λm+r ; q µ ) d (q λ ; q λ ) d n (q µ+λd+r ; q µ ) d For the sum on d in the last expression, the q-zeilberger algorithm evaluates it and gives us 0 for m n For m = n, it is easy: (q λ ; q λ ) n (q λ ; q λ ) n i λn q λn ( ) n i λn q λn ( )n q λn + λn λn (q µ+λn+r ; q µ ) n (q µ+λn+r ; q µ ) n (q µ+λn+r ; q µ ) n (q µ+λn+r ; q µ ) n = In that case, the equality is valid as well and so the proof is complete Now we present a proof for M We start with an introductory remark For all the identities that we need to prove, experiments indicate that they are Gosper-summable However, the entries

ASYMMETRIC GENERALIZATIONS OF THE FILBERT MATRIX 79 that we encounter in our instances, do not qualify for the q-zeilberger algorithm that we used in our earlier papers Therefore, it was necessary to guess the relevant quantities; the justification is then complete routine However, this guessing procedure is (with all the parameters involved) extremely time consuming, and so we confined ourselves to the demonstration of one such proof We hope that extensions of the q-zeilberger algorithm will be developed that fit our needs We deal now with L m,d L d,n n d m and prove that it is for n = m (there is only one term in the sum) and 0 for n > m since we have lower triangular matrices So let us assume m > n We will prove a general formula depending on an extra variable K: L m,d L d,n = ( )K n q λ K(K+) λkn+λ n(n ) n d K K(K+) q +r+sk s+(m n)λ K(K+) q +r+sk s (q λ ; q λ ) m (q s+λn+µ ; q µ ) K q λm λn (q λ ; q λ ) K n (q λ ; q λ ) m K (q λ ; q λ ) n (q s+λm+µ ; q µ ) K The formula we need follows from setting K := m Note that the RHS of formula equals 0 when K = m > n because of the term (q λ ; q λ ) m K in the denominator of the second row The proof of the formula is by induction Clearly it is true for K = n, and the induction step amounts to show that which equals n d K L m,d L d,n + L m,k+l K+,n = n d K+ L m,d L d,n, K(K+) ( ) K n q λ K(K+) λkn+λ n(n ) q +sk s+r+(m n)λ + K(K+) q +sk s+r (q λ ; q λ ) m (q λ ; q λ ) K n (q λ ; q λ ) m K (q λ ; q λ ) n (q s+λn+µ ; q µ ) K (q s+λm+µ ; q µ ) K (q λ ; q λ ) m (q λ ; q λ ) K (q λ ; q λ ) m K (q s+λ(k+)+µ ; q µ ) K+ (q s+λm+µ ; q µ ) K+ q (K+)(K+) +λ(m K )+sk+r (K+)(K+) q +sk+r ( ) K+ n q λ K(K+) +λ n(n+) λn(k+) (q s+λn+µ ; q µ ) K (q s+λ(k+)+µ ; q µ ) K q λm λn λ+µ q λn+ (K+) + λ µ (K+)+r+s(K ) (q λ ; q λ ) K λ+µ q λ(k+)+ (K+) + λ µ (K+)+r+s(K ) (q λ ; q λ ) n (q λ ; q λ ) K+ n = ( ) K+ n q λ (K+)(K+) λn(k+)+λ n(n )

80 KILIÇ AND PRODINGER q (K+)(K+) +s(k+) s+r+(m n)λ (K+)(K+) q +s(k+) s+r q λm λn or (q λ ; q λ ) m (q λ ; q λ ) K n+ (q λ ; q λ ) m K (q λ ; q λ ) n (q s+λn+µ ; q µ ) K+ (q s+λm+µ ; q µ ) K+, K(K+) ( q +sk s+r+(m n)λ (K+)(K+) )( q +sk+r ) ( q λ(k n+) )( q s+λm+µ+µk ) + ( q s+λ(k+)+µ+µk (K+)(K+) ) ( q +λ(m K )+sk+r ) λ+µ λn+ ( q (K+) + λ µ (K+)+r+s(K ) )( q λm λn ) = q λ+kλ nλ (K+)(K+) ( q +s(k+) s+r+(m n)λ ) K(K+) ( q +sk s+r )( q λ(m K ) )( q s+λn+µ+µk ), which is a routine check Thus we have the claimed result References C Berg, Fibonacci numbers and orthogonal polynomials, Arab J Math Sci 7 (0), 75 88 M E H Ismail, One parameter generalizations of the Fibonacci and Lucas numbers, Fibonacci Quart 46/47 (008/009), 67 80 3 E Kılıç, H Prodinger, A generalized Filbert matrix, Fibonacci Quart 48() (00), 9 33 4, The q-pilbert matrix, Int J Comput Math 89(0) (0), 370 377 5, Variants of the Filbert matrix, Fibonacci Quart 5() (03), 53 6 6, The generalized Lilbert matrix, submitted 7, The generalized q-pilbert Matrix, Math Slovaca, to appear 8 P Paule, A Riese, A mathematica q-analogue of Zeilberger s algorithm based on an algebraically motivated approach to q-hypergeometric telescoping; in: Special functions, q-series and related topics, Fields Inst Commun 4 (997), 79 0 9 H Prodinger, A generalization of a Filbert matrix with 3 additional parameters, Trans Roy Soc South Africa 65 (00), 69 7 0 T Richardson, The Filbert matrix, Fibonacci Quart 39(3) (00), 68 75 Department of Mathematics (Received 8 06 03) TOBB University of Economics and Technology Ankara 06560, Turkey ekilic@etuedutr Department of Mathematics University of Stellenbosch Stellenbosch 760, South Africa hproding@sunacza