A GENERALIZATION OF A FILBERT MATRIX WITH 3 ADDITIONAL PARAMETERS

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1 A GENERALIZATION OF A FILBERT MATRIX WITH 3 ADDITIONAL PARAMETERS HELMUT PRODINGER Abstract A generalize Filbert matrix from [] is further generalize, introucing 3 aitional parameters Explicit formulæ are erive for the LU-ecomposition, their inverses, an the Cholesky factorization The approach is to use q-analysis; the necessary ientities are clarifie on examples an can otherwise be left to automatic proofs (q- Zeilberger algorithm) Introuction In [], a generalize Filbert matrix with entries F i+j+r, where r is an integer parameter, was stuie (F n is the n-th Fibonacci number) The size of the matrix oes not really matter, an we can think about an infinite matrix F an restrict it whenever necessary to the first n rows resp columns an write F n See [] for historic remarks an pointers to the earlier literature Throughout, we will use the Binet form F n = αn β n α β qn = αn q, with q = β/α = α, so that α = i/ q In the previous paper, explicit formulæ where given for: The LU-ecomposition F = L U The inverse matrices L, U The inverse of the matrix F n The Cholesky ecomposition F = C C T with a lower triangular matrix C All the ientities hol for general q, an results about Fibonacci numbers come out as corollaries for the special choice of q Henceforth, we on t mention Fibonacci numbers any further In this paper, we generalize all this by introucing three extra parameters: x, y, an λ: F i,j := xi y j F λ(i+j)+r The previous instances come out as the special cases x = y = λ = (For the Cholesky case, we have to naturally assume that x = y) 000 Mathematics Subject Classification B39 Key wors an phrases Filbert matrix, Fibonacci numbers, q-analogues, LU-ecomposition, Cholesky ecomposition, Zeilberger s algorithm

2 HELMUT PRODINGER The main effort in obtaining these results is to guess the correct formulæ, which is oable with some experience, patience, an a computer The proofs are then routine, an in our previous paper, we only mentione that the q-zeilberger algorithm can o it Here, we want to be a bit more explicit, for the convenience of the reaer Throughout this paper we will use the following notations: (x; q) n = ( x)( xq) ( xq n ) an the Gaussian q-binomial coefficients [ n k ] = (q; q) n (q; q) k (q; q) n k Sometimes, when the parameter q is unerstoo, instea of (x; q) n, only (x) n is written Results Theorem The LU ecomposition is given by an L n, = i λ( n) q λ(n ) (q λ+r ; q λ ) (q λ+r ; q λ ) n (q λ+r ; q λ ) n+ (q λ+r ; q λ ) (q λ ; q λ ) n (q λ ; q λ ) n (q λ ; q λ ) y n U,n = (qλ ; q λ ) (q λ+r ; q λ ) (q λ+r ; q λ ) n (q λ ; q λ ) n ( q) (q λ+r ; q λ ) (q λ+r, q λ ) n+ (q λ ; q λ ) n ( q λn ) xn y for n ; for n <, these numbers are 0 q λ(n ) +λ +r r+ i λ(n+)+r+ ( ) λ(n+)+r Theorem The inverse matrices are given by L n, (n ) = qλ i λ(n+) ( ) (λ+)n+ (q λ+r ; q λ ) n+ (q λ ; q λ ) n (q λ+r ; q λ ) n y n (q λ+r ; q λ ) (q λ ; q λ ) n (q λ ; q λ ) (q λ+r ; q λ ) n an U,n = (q λ+r ; q λ ) n (q λ+r ; q λ ) n+ (q λ ; q λ ) (q λ ; q λ ) n (q λ ; q λ ) n (q λ+r ; q λ ) n (q λ+r ; q λ ) q x y n q λn + λ + r+ λn rn i λ(n+)+r ( ) n+ for n ; for n <, these numbers are 0 Theorem 3 The inverse matrix: (F n ) i,j = q λi +λj +r+ λ(i+j)n rn i λi+λj+r+ ( ) i+j+ ( q r+λi+λj )( q) x i y j (q λ+r ; q λ ) n+i (q λ+r ; q λ ) n+j (q λ ; q λ ) i (q λ ; q λ ) n i (q λ ; q λ ) j (q λ ; q λ ) n j (q λ+r ; q λ ) i (q λ+r ; q λ ) j

3 GENERALIZATION OF FILBERT MATRIX 3 Theorem 4 The Cholesky ecomposition is for x = y given by C n, = ( ) n r+ λn λn+r+ i q r+ 4 + λ( ) + r ( q λ+r )( q) (qλ ; q λ ) n (q λ+r ; q λ ) n x n (q λ ; q λ ) n (q λ+r ; q λ ) n+ This hols for n ; otherwise, these numbers are 0 3 How the prove the formulæ Let us consier one typical case: L m, L,n = i λ( m) q λ(m ) (q λ+r ; q λ ) (q λ+r ; q λ ) m (q λ ; q λ ) m y m (q λ+r ; q λ ) m+ (q λ+r ; q λ ) (q λ ; q λ ) m (q λ ; q λ ) q λ (n ) i λ(n+) ( ) (λ+)+n (q λ+r ; q λ ) n+ (q λ ; q λ ) (q λ+r ; q λ ) (q λ+r ; q λ ) n (q λ ; q λ ) n (q λ ; q λ ) n (q λ+r ; q λ ) y n = q λm + λn y m n ( ) n i λ(n m) (q λ+r ; q λ ) m (q λ ; q λ ) m (q λ ; q λ ) n (q λ+r ; q λ ) n q λ λ λn ( ) q r+λ (q λ+r ; q λ ) n+ (q λ+r ; q λ ) m+ (q λ ; q λ ) m (q λ ; q λ ) n It is clear that, since the matrices are of lower triangular type, we have to prove that the expression is 0 for m > n (For m = n, the entry is, which is not har to see, since it is only one term, not a sum) It is beneficial, for the evaluation of the sum, to switch to more convenient letters First, set Q = q λ an a = q r : Q ( ) n Now, we write q for Q an k for : m k=n ( ) (aq; Q) m+ aq (Q; Q) m (aq; Q) n+ (Q; Q) n q (k ) kn ( ) k ( aqk )(aq) n+k (aq) m+k (q) m k (q) k n Changing k to k + n (an introucing an irrelevant factor): m n q (k ) ( ) k ( aqk+n )(aq) n+k (aq) m+n+k (q) m n k (q) k Changing m to m + n (an introucing an irrelevant factor): m q (k ) ( ) k aqk+n m (aq n+k ) m+ k Eventually, we replace aq n by b: m q (k ) ( ) k bqk m (bq k ) m+ k

4 4 HELMUT PRODINGER We must show that this is 0 whenever m > 0 Now set F (m, k) = q (k ) ( ) k bqk m, (bq k ) m+ k G(m, k) = q (k+ ) ( ) k m, (bq k+ ) m k then it is trivial to check that F (m, k) = G(m, k) G(m, k ); this is the representation that the q-zeilberger algorithm provies Summing this over all k gives the result 0, as the righthan sie telescopes Alternatively, one can notice that the sum of interest is given by G(m, m), which evaluates to 0 Let us consier another case: m n U m, U,n = m n (q λ ; q λ ) m (q λ+r ; q λ ) m (q λ+r ; q λ ) (q λ ; q λ ) ( q) (q λ+r ; q λ ) m (q λ+r, q λ ) +m (q λ ; q λ ) m ( q λ ) x y m q λ( m) +λm +rm r+ i λ(+m)+r+ ( ) λ(+m)+r (q λ+r ; q λ ) n (q λ+r ; q λ ) n+ (q λ ; q λ ) (q λ ; q λ ) n (q λ ; q λ ) n (q λ+r ; q λ ) n (q λ+r ; q λ ) q x y n q λn + λ + r+ λn rn i λ(n+)+r ( ) n+ = (qλ ; q λ ) m (q λ+r ; q λ ) m (q λ+r ; q λ ) n y m n q λm (q λ+r ; q λ ) m (q λ ; q λ ) n (q λ+r ; q λ +λm +rm λn rn i λ(m+n) ( ) λm+n ) n m n ( ) q λ + λ λn (q λ+r ; q λ ) n+ (q λ+r ; q λ ) +m (q λ ; q λ ) m (q λ ; q λ ) n For the sum, let us write Q = q λ an a = q r : m n ( ) q + n (aq; Q) n+ (aq; Q) +m (Q; Q) m (Q; Q) n We must show that it is zero for m < n Let us replace by +m, ignore irrelevant factors an write q again for Q for convenience: Now replace n by n + m: 0 n m 0 n ( ) q + +m n (aq) n++m (aq) +m (q) (q) n m ( ) q + n (aq) n++m (aq) +m (q) (q) n

5 Ignore irrelevant factors: GENERALIZATION OF FILBERT MATRIX 5 0 n ( ) q + n n (aq ++m ) n Now write b = aq +m an the more traitional k instea of : n ( ) k q k + k n kn (bq k ) n k As before, call the term in the sum F (n, k) Then F (n, k) = G(n, k) G(n, k ), with G(n, k) = ( ) k q k + k kn (bq k n ) n bq n k Again, upon summing on k, we get 0, because of the telescoping property Other proofs follow the same pattern References [] E Kilic an H Proinger, A generalize Filbert matrix, Fibonacci Quart 48 (00), 9 33 Department of Mathematics, University of Stellenbosch 760 Stellenbosch South Africa aress: hproing@sunacza