Det-CUBO.dvi CUBO A Mathematical Journal Vol.12, No¯ 03, (1–12). October 2010 Partial Fractions and q-Binomial Determinant Identities WENCHANG CHU Dipartimento di Matematica, Università del Salento, Lecce-Arnesano P. O. Box 193, Lecce 73100, Italy email: chu.wenchang@unile.it CHENYING WANG College of Mathematics and Physics, Nanjing University of Information Science and Technology Nanjing 210044, P. R. China email: wang.chenying@163.com AND WENLONG ZHANG Department of Applied Mathematics, Dalian University of Technology, Dalian 116023, P. R. China email: wenlong.dlut@yahoo.com.cn ABSTRACT Partial fraction decomposition method is applied to evaluate a general determinant of shifted factorial fractions, which contains several Gaussian binomial determinant iden- tities. 2 Wenchang Chu, Chenying Wang & Wenlong Zhang CUBO 12, 3 (2010) RESUMEN El método de descomposición en fracción parciales aplicado para evaluar un determinante general de fracciones factoriales trasladadas, la cual contiene varias identidades determi- nante binomial Gaussiano. Key words and phrases: The Cauchy double alternant, Partial fractions, q-Binomial coeffi- cients. Math. Subj. Class.: 15A15, 11C20. Binomial determinant evaluation plays an important role in combinatorial enumeration, particularly in plane partitions. This paper will establish a very general determinant identity through partial fraction decomposition method. It will be shown to be useful in q-binomial determinant evaluations with several interesting known and new formulae being exemplified. 1 Partial Fraction Decomposition For two sequences {αk ,γk}k≥0, define the generalized shifted factorials by (x|α)0 = 1 and (x|α)n= n−1 ∏ k=0 (1 − xαk) with n ∈ N, (1a) ( y|γ)0 = 1 and ( y|γ)n = n−1 ∏ k=0 (1 − yγk) with n ∈ N. (1b) When αk = γk = q k for k ∈ N0, they will reduce to the usual shifted factorials (x; q)0 = 1 and (x; q)n = (1 − x)(1 − qx)···(1 − q n−1 x) with n ∈ N. (2) For the triangular matrix given by α=[αi j ]0≤i≤ j<∞, denote its j-th column by αj =(α0 j ,α1 j ,α2 j , ··· ,αj j ). Then the main result may be stated as follows. Theorem 1 (Generalized Cauchy determinant). Let {xk} n k=0 be distinct complex numbers. Then there holds the following determinant identity: det 0≤i, j≤n [ (xi|αj ) j (xi|γ) j+1 ] = ∏ 0≤i< j≤n(xi − x j )(αi j −γ j) ∏ n k=0 (xk|γ)n+1 . The very special case of this theorem with αi j = γi for i, j ∈ N0 results in the celebrated Cauchy’s double alternant (cf. [6, 7]): det 0≤i, j≤n [ 1 1 − xiγ j ] = ∏ 0≤i< j≤n(xi − x j )(γi −γ j) ∏ 0≤i, j≤n(1 − xiγ j ) . (3) CUBO 12, 3 (2010) Partial Fractions and q-Binomial Determinants 3 Proof. Expanding the rational function in partial fractions, we have (xi|αj ) j (xi|γ) j+1 = ∏ j−1 ι=0 (1 − xiαι j ) ∏ j k=0 (1 − xiγk) = j ∑ k=0 wk j 1 − xiγk where the connected coefficients are determined by the following limit relation wk j = lim xi → 1 γk (1 − xiγk) (xi|αj ) j (xi|γ) j+1 = ∏ j−1 ι=0 (αι j −γk) ∏ j ι=0: ι 6=k (γι −γk) . This leads us to the following determinant factorization det 0≤i, j≤n [ (xi|αj ) j (xi|γ) j+1 ] = det 0≤i,k≤n [ 1 1 − xiγk ] × det 0≤k, j≤n [ wk j ] . For the matrix [ wk j ] 0≤k, j≤n is upper triangular, its determinant is equal to the product of its diagonal entries: det 0≤k, j≤n [ wk j ] = n ∏ j=0 w j j = ∏ 0≤i< j≤n αi j −γ j γi −γ j . While the first determinant can be evaluated by Cauchy’s double alternant (2). Their combi- nation yields the determinant identity stated in Theorem 1. Shifting the γ-parameters by γk → γk−1, we may state the determinant identity in Theo- rem 1 in the following more convenient form. Proposition 2 (Determinant identity). Let {xk} n k=0 be distinct complex numbers. Then there holds the following determinant identity: det 0≤i, j≤n [ (xi|αj ) j (xi|γ) j ] = ∏ 0≤i< j≤n(xi − x j )(αi j −γ j−1) ∏n k=0 (xk|γ)n . Letting αi j = p i yj and γk = q k further in Proposition 2, we have the identity. Corollary 3 (Bibasic determinant evaluation formula). det 0≤i, j≤n [ (xi yj ; p) j (xi; q) j ] = q 2(n+13 ) ∏ 0≤i< j≤n (x j − xi ) n ∏ k=0 (q1−k yk; p)k (xk; q)n . From this corollary, we can derive numerous q-binomial determinant identities. 2 q-Binomial Determinant Identities Define the Gaussian binomial coefficients by [ x n ] = (q1+x−n; q)n (q; q)n where n ∈ N0 and x ∈ C. Applying Corollary 3, we show now ten classes of q-binomial determinant identities. 4 Wenchang Chu, Chenying Wang & Wenlong Zhang CUBO 12, 3 (2010) 2.1 Expressing the q-binomial coefficient in terms of shifted factorials [ X i − j A ] = q −A j [ X i A ] (qA−X i ; q) j (q−X i ; q) j we derive the corresponding determinant formula det 0≤i, j≤n [ [ X i − j A ] ] = ∏ 0≤i< j≤n (q−X j − q−X i )(1 − q1+A+i− j ) (4a) × q2( n+1 3 )−A( n+1 2 ) (q; q)n+1n n ∏ k=0 [ X k A ][ n − 1 − X k n ]−1 . (4b) 2.2 Rewriting the q-binomial coefficient in terms of shifted factorials [ A X i − j ] = (−1) j q−( j 2)+ j X i [ A X i ] (q−X i ; q) j (q1+A−X i ; q) j we get the corresponding determinant identity det 0≤i, j≤n [ q − j X i [ A X i − j ] ] = ∏ 0≤i< j≤n (q−X i − q−X j )(1 − q−A+i− j ) (5a) × q( n+1 3 )+(1+A)( n+1 2 ) (q; q)n+1n n ∏ k=0 [ A X k ][ A + n − X k n ]−1 . (5b) 2.3 Reformulating the q-binomial coefficient in terms of shifted factorials [ A + X i − j X i − j ] = q −A j [ A + X i A ] (q−X i ; q) j (q−A−X i ; q) j we obtain the following determinant evaluation formula det 0≤i, j≤n [ [ A + X i − j X i − j ] ] = ∏ 0≤i< j≤n (q−X j − q−X i )(1 − q1+A+i− j ) (6a) × q2( n+1 3 )−2A( n+1 2 ) (q; q)n+1n n ∏ k=0 [ A + X k A ][ −1 − A + n − X k n ]−1 . (6b) 2.4 Applying the q-binomial relation [ X i + j A ] = [ X i A ] (q1+X i ; q) j (q1−A+X i ; q) j CUBO 12, 3 (2010) Partial Fractions and q-Binomial Determinants 5 we find the corresponding determinant formula det 0≤i, j≤n [ [ X i + j A ] ] = ∏ 0≤i< j≤n (qX j − qX i )(1 − q1+A+i− j ) (7a) × q2( n+1 3 )+(1−A)( n+1 2 ) (q; q)n+1n n ∏ k=0 [ X k A ][ X k − A + n n ]−1 . (7b) 2.5 Observing the q-binomial relation [ A X i + j ] = (−1) j q(A−X i ) j−( j 2) [ A X i ] (q−A+X i ; q) j (q1+X i ; q) j we recover the determinant identity due to Carlitz [4] (cf. Chu [5] also) det 0≤i, j≤n [ q j X i [ A X i + j ] ] = ∏ 0≤i< j≤n (qX i − qX j )(1 − q−A+i− j ) (8a) × q( n+1 3 )+(1+A)( n+1 2 ) (q; q)n+1n n ∏ k=0 [ A X k ][ X k + n n ]−1 . (8b) 2.6 By invoking the q-binomial relation [ A + X i + j X i + j ] = [ A + X i A ] (q1+A+X i ; q) j (q1+X i ; q) j we recover another determinant identity due to Carlitz [4] (see Menon [9] also) det 0≤i, j≤n [ [ A + X i + j X i + j ] ] = ∏ 0≤i< j≤n (qX j − qX i )(1 − q1+A+i− j ) (9a) × q2( n+1 3 )+( n+1 2 ) (q; q)n+1n n ∏ k=0 [ A + X k A ][ X k + n n ]−1 (9b) which reduces, for q → 1, to the binomial determinant of Ostrowski [10]. Furthermore for δ = 0, 1, we can show the following determinant identity det 0≤i, j≤n [ C (δ) X i + j (q) ] = (2q)(1+n)(1+n+δ)+2 ∑n ι=0 Xι q n(n+1)(1+2n+6δ)/6 (10a) × n ∏ k=0 (q; q2)1+k(q; q 2 )δ+X k (q2; q2)1+δ+X k+n ∏ 0≤i< j≤n (q2X i − q2X j ) (10b) where the q-Catalan numbers due to Andrews [2] has been slightly extended by C (δ) n (q) := (2q)1+δ+2n 1 − q2+2δ+2n [ δ+ 2n n ] 1 − q (−q; q)n (−q; q)δ+n . (11) 6 Wenchang Chu, Chenying Wang & Wenlong Zhang CUBO 12, 3 (2010) When xk = k +ℓ, we get the following Hankel determinant identity det 0≤i, j≤n [ C (δ) i+ j+ℓ (q) ] = (2q)(1+n)(1+δ+2n+2ℓ)qn(n+1)(4n+6ℓ+6δ−1)/6 (12a) × n ∏ k=0 (q; q)1+2k (q; q 2)δ+k+ℓ (q2; q2)1+δ+k+n+ℓ . (12b) Letting δ = 0 and q → 1, we recover further the related results [1, 8, 11] on the classical Catalan numbers Cn = 1 n+1 (2n n ) : det 0≤i, j≤n [ Ci+ j ] = 1, det 0≤i, j≤n [ Ci+ j+1 ] = 1, det 0≤i, j≤n [ Ci+ j+2 ] = n + 2. (13) 2.7 By means of the q-binomial relation [ X i + Y j j ][ A + X i j ]−1 = q (Y j −A) j (q−X i−Y j ; q) j (q−A−X i ; q) j we get the following determinant identity det 0≤i, j≤n [ [ X i + Y j j ][ A + X i j ]−1] = q 2(n+13 )− ∑n k=0 (2k A+n X k−kYk ) (q; q)n+1n ∏ n k=0 [ n−1−A−X k n ] (14a) × ∏ 0≤i< j≤n (qX i − qX j )(1 − q1+A−Y j+i− j ). (14b) 2.8 In view of the q-binomial relation [ X i + Y j + j Y j ][ X i + Y j Y j ]−1 = (q1+X i+Y j ; q) j (q1+X i ; q) j we obtain the corresponding determinant formula det 0≤i, j≤n [ [ X i + Y j + j Y j ][ X i + Y j Y j ]−1] = q2( n+1 3 )+( n+1 2 ) (q; q)n+1n n ∏ k=0 [ X k + n n ]−1 (15a) × ∏ 0≤i< j≤n (qX j − qX i )(1 − q1+Y j +i− j ). (15b) 2.9 According to the q-binomial relation [ A + X i + Y j j ][ X i + j j ]−1 = (q1+A+X i +Y j− j ; q) j (q1+X i ; q) j CUBO 12, 3 (2010) Partial Fractions and q-Binomial Determinants 7 we derive the corresponding determinant identity det 0≤i, j≤n [ [ A + X i + Y j j ][ X i + j j ]−1] = q2( n+1 3 )+( n+1 2 ) (q; q)n+1n n ∏ k=0 [ X k + n n ]−1 (16a) × ∏ 0≤i< j≤n (qX j − qX i )(1 − q1+A+Y j +i−2 j ). (16b) 2.10 Similarly, the q-binomial relation [ X i + Y j j ][ A + X i − j n − j ] = q (Y j −A) j [ n j ][ A + X i n ] (q−X i −Y j ; q) j (q−A−X i ; q) j leads us to the following binomial determinant evaluation formulae det 0≤i, j≤n [ [X i+Y j j ][ A+X i − j n− j ] ] = ∏ 0≤i< j≤n (q−X j − q−X i )(1 − q1+A+i− j−Y j ) (17a) × q ∑n k=0 (k−1−2A+Yk)k (q; q)n+1n n ∏ k=0 [ n k ][ A+X k n ] [ −1−A+n−X k n ] , (17b) det 0≤i, j≤n [ [ X i+ j j ][A+X i +Y j n− j ] ] = ∏ 0≤i< j≤n (q−X i − q−X j )(1 − q1+n−A−Yn− j +i− j ) (18a) × q ∑ n k=0 (k−1+A−2n+Yn−k)k (q; q)n+1n n ∏ k=0 [ n k ][ n+X k n ] [ −1−X k n ] ; (18b) where the last identity is derived from the first one under substitution j → n− j on the column index. 3 Duplicate Determinant Identities Performing the parameter replacements in Proposition 2 xk → axk + c/xk , γk → dγk/(1 + acd 2 γ 2 k ), αi j → bαi j /(1 + ab 2 cα 2 i j ); 8 Wenchang Chu, Chenying Wang & Wenlong Zhang CUBO 12, 3 (2010) and then applying factorizations xi − x j → (xi − x j )(a − c/xi x j ), αi j −γk → (bαi j − dγk)(1 − abcdαi j γk) (1 + ab2 cα2 i j )(1 + acd2γ2 k ) , 1 − xiγk → (1 − adγk xi )(1 − cdγk /xi ) 1 + acd2γ2 k , 1 − xkαi j → (1 − abxkαi j )(1 − bcαi j /xk) 1 + ab2 cα2 i j ; we find the following duplicate determinant identity. Proposition 4. Let {xk} n k=0 be distinct complex numbers. Then there holds the following de- terminant identity: det 0≤i, j≤n [ (abxi|αj ) j(bc/xi |αj ) j (adxi|γ) j(cd/xi |γ) j ] = ∏ 0≤i< j≤n (bαi j − dγ j−1)(1 − abcdαi j γ j−1) × ∏ 0≤i< j≤n(xi − x j )(a − c/xi x j) ∏ n k=0 (adxk|γ)n(cd/xk|γ)n . This identity contains the following three determinant evaluations. Corollary 5 (a = b = 1 and γk → 0 in Proposition 4). det 0≤i, j≤n [ (xi|αj ) j (c/xi|αj ) j ] = ∏ 0≤i< j≤n { αi j (xi − x j )(1 − c/xi x j ) } . Corollary 6 (d = 1 and αi j → 0 in Proposition 4). det 0≤i, j≤n [ 1 (axi|γ) j (c/xi|γ) j ] = ∏ 0≤i< j≤n(x j − xi )(a − c/xi x j) ∏n k=0 (axk|γ)n(c/xk|γ)n n ∏ ℓ=1 γ ℓ ℓ−1. Putting αi j = p i yj and γk = q k in Proposition 4, we find the following determinant eval- uation formula of factorial fractions with two different bases. Corollary 7 (Bibasic determinant identity). det 0≤i, j≤n [ (abxi yj ; p) j (bc yj /xi ; p) j (adxi ; q) j (cd/xi ; q) j ] = d( n+1 2 ) ∏ 0≤i< j≤n (x j − xi )(a − c/xi x j ) × q 2(n+13 ) n ∏ k=0 (q1−k b yk /d; p)k (q k−1 abcd yk ; p)k (adxk ; q)n (cd/xk ; q)n . CUBO 12, 3 (2010) Partial Fractions and q-Binomial Determinants 9 When p = q and yk = 1, it reduces to the following determinant identity det 0≤i, j≤n [ (abxi ; q) j (bc/xi ; q) j (adxi ; q) j (cd/xi ; q) j ] = b( n+1 2 ) ∏ 0≤i< j≤n (xi − x j )(a − c/xi x j ) (19a) × q( n+1 3 ) n ∏ k=0 (d/b; q)k (q k−1abcd; q)k (adxk ; q)n(cd/xk ; q)n . (19b) The determinant evaluation formulae established in this section contain numerous q-binomial determinant identities as special cases, which will be illustrated by the following five exam- ples. 3.1 Expressing the q-binomial coefficients in terms of shifted factorials [ X i +A j ][ X i −B−C n− j ] [ X i +B j ][ X i −A−C n− j ] = q (A−B) j [ X i −B−C n ] [ X i −A−C n ] × (q1+X i−A−C−n; q) j (q −X i−A ; q) j (q1+X i−B−C−n; q) j (q−X i−B; q) j we establish from Corollary 7 the determinant evaluation formula det 0≤i, j≤n [ [ X i +A j ][ X i −B−C n− j ] [ X i +B j ][ X i −A−C n− j ] q( j 2) ] = ∏ 0≤i< j≤n (qX i − qX j )(1 − qn−1+C−X i−X j ) (20a) × qB( n+1 2 ) (q; q)n+1n n ∏ k=0 [ n+A+B+C−k k ][ B−A k ][ X k−B−C n ] [ n k ][ X k+B n ][ n−1+B+C−X k n ][ X k−A−C n ] (20b) which contains, as special case, the following q-binomial determinant identity det 0≤i, j≤n [ [ λi+A j ][ λi−B n− j ] [ λi+B j ][ λi−A n− j ] q( j 2) ] = ∏ 0≤i< j≤n (qλi − qλ j )(1 − qn−1−λi−λ j) (21a) × qB( n+1 2 ) (q; q)n+1n n ∏ k=0 [ n+A+B−k k ][ B−A k ][ λk−B n ] [ n k ][ λk+B n ][ n−1+B−λk n ][ λk−A n ] . (21b) 3.2 Rewriting the q-binomial coefficients in terms of shifted factorials [X i+Y j j ][A−X i+Y j j ] [ B+X i j ][ A+B−X i j ] = q 2 j(Y j−B) (q−X i −Y j ; q) j (q X i −A−Y j ; q) j (q−X i −B; q) j (qX i −A−B; q) j we recover from Corollary 7 the determinant identity due to Joris Van Jeugt det 0≤i, j≤n   [X i+Y j j ][A−X i +Y j j ] [ B+X i j ][ A+B−X i j ] q( j 2)   = ∏ 0≤i< j≤n (qX i − qX j )(1 − qA−X i −X j ) (22a) × q ∑n k=0 kYk (q; q)n+1n n ∏ k=0 [1+A+B+Yk−k k ][ B−Yk k ] [ n k ][ B+X k n ][ A+B−X k n ] . (22b) 10 Wenchang Chu, Chenying Wang & Wenlong Zhang CUBO 12, 3 (2010) This identity can further be specialized to the q-binomial determinant evaluation det 0≤i, j≤n   [ A+λi+ j j ][ A−λi+ j j ] [ B+λi j ][ B−λi j ] q( j 2)   = ∏ 0≤i< j≤n (q−λi − q−λ j )(1 − qλi+λ j ) (23a) × q ∑n k=0 k(A+k) (q; q)n+1n n ∏ k=0 [1+A+B k ][ B−A−k k ] [ n k ][ B+λk n ][ B−λk n ] . (23b) 3.3 Reformulating the q-binomial coefficients in terms of shifted factorials [ X i +Y j+ j X i −Y j−A− j ] [ X i+Y j X i−Y j −A ][ X i +C+ j X i −A−C− j ] = q (C−Y j ) j [ A+2C+2 j 2C−2Y j ] [ A+2C A+2Y j ][ C+X i A+2C ] × (q1+X i+Y j ; q) j (q A−X i +Y j ; q) j (q1+X i+C ; q) j (qA−X i +C; q) j we derive from Corollary 7 the following determinant formula det 0≤i, j≤n [ [ X i +Y j+ j X i −Y j−A− j ] [ X i+Y j X i−Y j −A ][ X i +C+ j X i −A−C− j ] ] = ∏ 0≤i< j≤n (qX i − qX j )(1 − qA−1−X i −X j ) (24a) × q 2(n+23 )+( n+1 2 )(C−A)+n ∑n k=0 X k (q; q)n+1n [ A+2C+2n 2n ]n+1[2n n ]n+1 n ∏ k=0 [ A+2C+2k 2C−2Yk ][ Yk−C k ][ −k−A−C−Yk k ] [ n k ][ A+2C A+2Yk ][ X k+C+n A+2C+2n ] (24b) which reduces, for X i = bi and Y j = 0, to the q-binomial determinant identity: det 0≤i, j≤n [ [ bi + j 2 j ][ bi + c + j 2c + 2 j ]−1] = ∏ 0≤i< j≤n (q−bi − q−b j )(1 − q1+bi+b j ) (25a) × q ∑n k=0 k(nb+c+k) (q; q)n+1n [2c+2n 2n ]n+1[2n n ]n+1 n ∏ k=0 [2c+2k 2c ][ −c k ][ −k−c k ] [ n k ][ bk+c+n 2c+2n ] . (25b) 3.4 According to Corollary 6, the q-binomial relation q j X i [ X i + A − j X i + j ] = (−1) j q( j 2)−A j (q; q)X i +A (q; q)X i (q; q)A−2 j × 1 (q1+X i ; q) j (q−A−X i ; q) j yields the determinant evaluation formula det 0≤i, j≤n [ q j X i [ X i + A − j X i + j ] ] = ∏ 0≤i< j≤n (q−X i − q−X j )(1 − q1+A+X i+X j ) (26a) × n ∏ k=0 [ X k + A − n X k + n ] qn X k (q1+A−2n; q)2k . (26b) CUBO 12, 3 (2010) Partial Fractions and q-Binomial Determinants 11 In particular for X i = c + bi, it becomes the q-binomial determinant identity det 0≤i, j≤n [ q bi j [ a + bi − j c + bi + j ] ] ∏ 0≤i< j≤n (q−bi − q−b j )(1 − q1+a+c+bi+b j) (27a) × q nb(n+12 ) n ∏ k=0 (q; q)a+bk−n (q; q)a−c−2k (q; q)c+bk+n (27b) which is the q-analogue of the determinant evaluated by Amdeberhan and Zeilberger [3, Eq 2]. 3.5 In view of Corollary 5, the q-binomial relation q − j X i [ X i + Y j + j X i − Y j + A − j ][ X i − Y j + A X i + Y j ] = (−1) j q(A−Y j ) j−( j 2) (q; q)2Y j −A+2 j(q; q)A−2Y j × (q1+X i +Y j ; q) j (q Y j −X i−A ; q) j leads to the following determinant evaluation det 0≤i, j≤n [ q − j X i [ X i + Y j + j X i − Y j + A − j ][ X i − Y j + A X i + Y j ] ] (28a) = ∏ 0≤i< j≤n(q −X j − q−X i )(1 − q1+A+X i +X j ) ∏n k=0 (q; q)A−2Yk (q; q)2Yk−A+2k . (28b) In particular for X i = a + bi and Y j = 0, the last identity gives det 0≤i, j≤n [ q −bi j [ a + bi + j c + bi − j ] ] = ∏ 0≤i< j≤n(q −b j − q−bi )(1 − q1+a+c+bi+b j) ∏n k=0 (q1+a+bk; q)c−a (q; q)a−c+2k (29) which results in the q-analogue of the binomial determinant identity due to Amdeberhan and Zeilberger [3, Eq 1]. Similarly, letting xi = q a+bi and αi j = q d j−i, we find from Corollary 5 another determi- nant identity det 0≤i, j≤n [ [ a + bi + d j j ][ c − bi + d j j ] q( j 2) ] (30a) = n ∏ k=0 qk(c+d k) (q; q)2 k ∏ 0≤i< j≤n (q−bi − q−b j )(1 − qa−c+bi+b j ) (30b) which is the q-analogue of the result in Amdeberhan and Zeilberger [3, Eq 14]. The list of examples can be endless. However, we are not going further to prolong it due to the space limitation. 12 Wenchang Chu, Chenying Wang & Wenlong Zhang CUBO 12, 3 (2010) References [1] AIGNER, M., Catalan-like numbers and determinants, J. Combin. Theory (Ser. A) 87 (1999), 33–51. 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