A QUADRATIC FORMULA FOR BASIC HYPERGEOMETRIC SERIES RELATED TO ASKEY-WILSON POLYNOMIALS

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00 Number 0 Pages S XX) A QUADRATIC FORMULA FOR BASIC HYPERGEOMETRIC SERIES RELATED TO ASKEY-WILSON POLYNOMIALS VICTOR J W GUO MASAO ISHIKAWA HIROYUKI TAGAWA AND JIANG ZENG Communicated by Sergei Suslov) Dedicated to Srinivasa Ramanujan on the occasion of his 125th birth anniversary Abstract We prove a general quadratic formula for basic hypergeometric series from which simple proofs of several recent determinant and Pfaffian formulas are obtained A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials We also show how to derive a recent double-sum formula for the moments of Askey- Wilson polynomials from Newton s interpolation formula 1 Introduction Throughout this paper we assume that q is a fixed number in 0 1) A q-shifted factorial is defined by a; q) = 1 aq k ) and a; q) n = a; q) aq n for n Z ; q) Following Gasper and Rhaman 7 we shall use the abbreviated notation a 1 a 2 a r ; q) n = a 1 ; q) n a 2 ; q) n a r ; q) n for n Z A basic hypergeometric series with r numerators and s denominators is then defined by a1 a 2 a r a 1 a 2 a r ; q) n rφ s ; q z = 1) n q 2) ) 1+s r b 1 b n z n s q b 1 b s ; q) n n=0 The Askey-Wilson polynomials p n x; a b c d; q) n N) are the 4 φ polynomials 7 ab ac ad; q) n q a n 4φ n abcdq ae Iθ ae Iθ ; q q ab ac ad where x = cos θ and I is a complex number such that I 2 = 1 Received by the editors November and in revised form July Mathematics Subject Classification Primary C45; D45; Secondary 05A19 Key words and phrases quadratic formula of basic hypergeometric series Askey-Wilson polynomials moments of Askey-Wilson polynomials Gram determinants Pfaffians Desnanot Jacobi adjoint matrix theorem Partially supported by JSPS Grant-in-Aid for Scientific Research C) Partially supported by CMIRA COOPERA 2012 de la Région Rhône-Alpes 1 c XXXX American Mathematical Society

2 2 V J W GUO M ISHIKAWA H TAGAWA AND J ZENG Our main result is the following quadratic formula for basic hypergeometric series which was discovered by applying Desnanot Jacobi adjoint matrix theorem to compute some determinants of Mehta-Wang type or deformed Gram determinants of orthogonal polynomials 20 Theorem 11 Let r s 0 a b c d q C E r = e 1 e 2 e r ) C r F s = f 1 f 2 f s ) C s Then we have 11) a b)a c)bc d)1 d) a r+4 φ 1 bc bcq 2 c dq 1 E r s+ aq 1 bq 1 bcd 1 ; q z F s = a d)1 b)1 c)bc ad) a r+4 φ 1 bc bcq 2 cq 1 d E r s+ aq 1 b bcd 1 q 1 ; q z F s r+4 φ s+ a 1 bcq 1 bcq 2 c d E r a bq 1 bcd 1 q 1 F s ; q z 1 a)b d)c d)a bc) a r+4φ 1 bc bc c dq E r q s+ aq bq bcd 1 F s q ; q qs r z r+4φ s+ a 1 bc bc cq d E r q aq b bcd 1 q F s q ; q qs r z r+4φ s+ a 1 bcq bc c d E r q a bq bcd 1 q F s q ; q qs r z Let s = r d = a 0 c = a 1 b = 0 a = b 1 e 1 = f 1 = 0 e i = a i and f j = b j 2 j r) we get the following result by shifting r to r Corollary 12 For r 1 there holds 12) a 0 1)a 1 b 1 ) a0 /q a r+1 φ 1 a 2 a r r ; q z b 1 /q b 2 b r = a 0 a 1 )1 b 1 ) a0 a r+1 φ 1 a 2 a r r ; q z b 1 b 2 b r 1 a 1 )a 0 b 1 ) a0 a r+1 φ 1 /q a 2 a r r ; q z b 1 /q b 2 b r a0 q a 1 a 2 q a r q r+1φ r b 1 q b 2 q b r q ; q z a0 a 1 a 2 q a r q r+1φ r b 1 b 2 q b r q ; q z a0 a 1 q a 2 q a r q r+1φ r b 1 q b 2 q b r q ; q z Taking r = z = q a 0 = q n+1 a 1 = abcdq a 2 = ae iθ a = ae iθ b 1 = abq b 2 = ac and b = ad in Corollary 12 we obtain the following quadratic relation for Askey-Wilson polynomials Corollary 1 Let n be a positive integer There holds 1) ab1 q )1 cdq n 2 )p n x; a b c d; q)p n 2 x; aq bq c d; q) = 1 abq )1 abcdq )p x; a b c d; q)p x; aq bq c d; q) 1 ab)1 abcdq 2n 2 )p x; aq b c d; q)p x; a bq c d; q) We shall prove Theorem 11 in the next section In Section we use the Desnanot-Jacobi adjoint matrix theorem and Corollary 1 to derive a determinant formula for Askey-Wilson polynomials cf Theorem 1) which turns out to be a generalization of several recent determinant and Pffafian evaluations in In Section 4 we connect the determinant formula in Theorem 1 to a Gram determinant formula of Askey-Wilson polynomials and show how to derive a

3 A QUADRATIC FORMULA FOR BASIC HYPERGEOMETRIC SERIES recent double sum formula for the moments of Askey-Wilson polynomials in 4 9 from Newton s interpolation formula For 0 k n we write 2 Proof of Theorem 11 A k = a b)a c)bc d)1 d)α k α n k+1 a 1 bc bcq 2 c dq 1 E r ; q) k a 1 bc bc c dq E r q; q) n k q aq 1 bq 1 bcd 1 F s ; q) k q aq bq bcd 1 F s q; q) n k B k = a d)1 b)1 c)bc ad)α k α n k+1 a 1 bc bcq 2 cq 1 d E r ; q) k a 1 bc bc cq d E r q; q) n k q aq 1 b bcd 1 q 1 F s ; q) k q aq b bcd 1 q F s q; q) n k C k = 1 a)b d)c d)a bc)α k α n k+1 a 1 bcq 1 bcq 2 c d E r ; q) k a 1 bcq bc c d E r q; q) n k q a bq 1 bcd 1 q 1 F s ; q) k q a bq bcd 1 q F s q; q) n k } s r where α k = { 1) k q kk 1) 2 for k 0 Equating the coefficients of z n on both sides of 11) yields the equivalent identity n 21) A k B k + C k ) = 0 The key point to prove 21) is the observation that 22) A k B k + C k + A n k+1 B n k+1 + C n k+1 ) = 0 for 0 k n + 1 where A n+1 = B n+1 = C n+1 = 0 Indeed summing 22) over k from 0 to n + 1 on both sides yields immediately 21) To prove 22) we start from the identity 2) a b)a c)d x)bc dx)x ay) x by)x cy)y dz)ax bcy)dy bcz) a d)b x)c x)ad bc)x ay) y bz)y cz)x dy)ax bcy)dx bcy) + b d)c d)a x)ax bc)y az) x by)x cy)x dy)ay bcz)dx bcy) = xya b)a c)a d)b d)c d) 1 y)ad bc)x bcz)x 2 bcy)y 2 xz) which can be easily checked either by hands or by Maple Replacing x y z) by q q k q n ) in 2) and multiplying both sides of the resulting identity by we obtain 24) c d a 1 bc; q) k 1 bcq 2 E r ; q) k c d a 1 bc bc E r q; q) n k aq 1 bq 1 bcd 1 q 1 ; q) k+1 q F s ; q) k aq bq bcd 1 q q F s q; q) n k A k B k + C k = q n k+1 q k )G k G n k+1 Ξ

4 4 V J W GUO M ISHIKAWA H TAGAWA AND J ZENG where and G k = 1 qk )1 bcq k 2 )a 1 bc c d; q) k 1 bc; q) k 2 E r ; q) k α k a b bcd 1 q; q) k F s ; q) k Ξ = a b)a c)a d)b d)c d)ad bc)1 bcq ) 1 a)1 b)1 bcd 1 )1 bcq 2 )1 bcq 1 )F s ; q) 1 adq 2 1 aq 1 )1 bq 1 )1 bcd 1 q 1 )E r ; q) 1 which is independent of k Clearly 24) implies 22) Application to determinant and Pfaffian evaluation Given a matrix M if i 1 i r resp j 1 j r ) are row resp column) indices we denote by M j1jr i 1i r the matrix that remains when the rows i 1 i r and columns j 1 j r are deleted Let n 2 and M be an n n matrix Then the Desnanot- Jacobi adjoint matrix theorem 1 Lemma 77 reads 1) det M det M 1 n 1 n = det M 1 1 det M n n det M 1 n det M n 1 where we set det M 1 n 1 n = 1 if n = 2 Theorem 1 For n 1 0 i n 1 and 1 j n let 2) Then ) where 4) B ij = ab; q) i+j 1 bq 1+j ) abcd; q) i+j c + d 2x + 1 cd)aq i + bq j 1 ) abc + d 2cdx)q i+j 1 detb ij ) 0 i 1 j n = D n a b) p n x; a b c d; q) D n a b) = a n)/2 b nn+1)/2 q n)2)/6 Proof By 4) we have Therefore 5) and 6) i=0 ab cd q; q) i abcd; q) n+i D n a b)/d a b) = a b n q ab cd; q) q; q) )2 abcdq ; q) n abcd; q) 2n 2 D n aq b)/d n a b) = q n)/2 abq; q) abcdq n ; q) n 1 abcd) n 1 ab) D n a b)/d a b) D aq bq)/d n 2 aq bq) = abab; q) 21 cdq n 2 )1 q ) 1 abq )1 abcdq )abcd; q) 2 D aq b)/d a b) D aq bq)/d a bq) = 1 abq 1 abcdq 2n 2 1 abq 1 abq 1 abcdq 1 ab ) n 2 1 abcd ) 1 abcdq

5 A QUADRATIC FORMULA FOR BASIC HYPERGEOMETRIC SERIES 5 Let M n a b) := detb ij ) 0 i 1 j n For n = 1 formula ) is obvious Assume that n 2 Applying 1) to the determinant in ) we obtain M n a b) M n 2 aq bq) = ab; q) 2 M a b) M aq bq) abcd; q) 2 1 ab ) n 1 abcdq ) n 2M aq b) M a bq) 1 abcd 1 abq It suffices to show that if we substitute M n a b) by D n a b)p n x; a b c d; q) the above identity still holds ie for n 2 7) D n a b)d n 2 aq bq)p n x; a b c d; q)p n 2 x; aq bq c d; q) = ab; q) 2 D a b)d aq bq)p x; a b c d; q)p x; aq bq c d; q) abcd; q) 2 1 ab ) n 1 abcdq ) n 2D aq b)d a bq) 1 abcd 1 abq p x; a bq c d; q)p x; aq b c d; q) Dividing the two sides of 7) by D a b)d aq bq) and applying the two identities 5) and 6) we see that 7) is exactly the quadratic formula 1) The result then follows by induction on n The following formula is an extension of Nishizawa s q-analogue of Mehta-Wang s formula Corollary 2 For n 1 there holds 8) det q i 1 cq j 1 ) aq; q) i+j 2 abq 2 ; q) i+j 2 )1 ij n n = 1) n a nn ) 2 q nn+1)2n 5) 6 abcq; q 2 q; q) k 1 aq; q) k bq; q) k 2 ) n abq 2 ; q) k+n 2 q 4 φ n abq n acq) 1 2 acq) 1 2 ; q q aq abcq) 1 2 abcq) 1 2 Proof Since the Askey-Wilson polynomials are symmetric on a and b in ) replacing p n x; a b c d; q) by p n x; b a c d; q) and making the following substitution: x 0 a aq/c) 1/2 I b acq) 1/2 I c b 1/2 I d b 1/2 I where I 2 = 1 gives 8) as bq; q) 1 = 1/1 b) If c = 1 we can sum the 4 φ in 8) by Andrews terminating q-analogue of Watson s formula 7 II17 0 if n is odd q 4φ n a 2 q n+1 b b aq aq b 2 ; q q = b n q a 2 q 2 /b 2 ; q 2 ) n/2 a 2 q 2 b 2 q; q 2 if n is even ) n/2 and deduce the following result which was first proved in 11 and is also a q- analogue of 14 Theorem 6

6 6 V J W GUO M ISHIKAWA H TAGAWA AND J ZENG Corollary For m 1 there holds det q i 1 q j 1 ) aq; q) i+j 2 abq 2 ; q) i+j 2 )1 ij 2m = a 2mm 1) q 2mm 1)4m+1) m q aq; q) 2k 1 bq; q) 2k 2 abq 2 ; q) 2k+m) Recall 1 that the Pfaffian of a skew-symmetric matrix A = A ij ) 1 ij 2m is defined by 9) PfA = sgn π A ij π M12m i<j ij matched in π Here Ma b is the set of perfect matchings of the complete graph on {a b} for any nonnegative integers a and b such that a < b and sgn π = 1) cr π where cr π is the number of matched pairs i j) and i j ) in π such that i < i < j < j The following result was first proved by Ishikawa et al 11 Corollary 4 For m 1 there holds Pf q i 1 q j 1 ) aq; q) ) i+j 2 abq 2 ; q) i+j 2 1 ij 2m = a mm 1) q mm 1)4m+1) m ) 2 q aq; q) 2k 1 bq; q) 2k 2 abq 2 ; q) 2k+m) Proof As the square of the Pfaffian of any skew-symmetric matrix is its determinant we derive from Corollary that 10) Pf q i 1 q j 1 ) aq; q) ) i+j 2 abq 2 ; q) i+j 2 1 ij 2m = ε m a mm 1) q mm 1)4m+1) m q aq; q) 2k 1 bq; q) 2k 2 abq 2 ; q) 2k+m) where ε 2 m = 1 By 10) the factor ε m is a rational function of a b and q and only takes values 1 or 1 Hence for fixed m we must have ε m = 1 or ε m = 1 regardless the values of a b and q It remains to show that ε m = 1 for all m 1 Obviously we have ε 1 = 1 Suppose that m 2 Taking b = 0 and replacing a by q a 1 the identity 10) reduces to 11) Pf q i q j )q a ; q) i+j )0 ij 2m 1 mm 1)4m+1) mm 1)a 1)+ = ε m q m q q a ; q) 2k 1 Using the q-gamma function 7 p 20 Γ q x) = q; q) q x ; q) 1 q) 1 x 0 < q < 1

7 A QUADRATIC FORMULA FOR BASIC HYPERGEOMETRIC SERIES 7 we can rewrite 11) as 12) Pf q i j i q Γ q a + i + j) ) 0 ij 2m 1 mm 1)4m+1) m mm 1)a 1)+ = ε m q 2k 1 q! Γ q a + 2k 1) where n q! = 1 q 2 q n q and k q = 1 q k )/1 q) If we multiply both sides of 12) by a + 1 q and let a tend to 1 then the right-hand side becomes 1) π M02m 1 ε m q mm 1)4m 5) m m 1 2k 1 q! Γ q 2k) On the other hand by 9) the left-hand side can be written as 14) sgn π lim a + 1q q i j i q Γ q a + i + j) a 1 i<j ijmached in π In this sum matchings π for which all matched pairs i j satisfy i + j > 1 will not contribute because the corresponding summands vanish Therefore the survival matchings must match 0 and 1 and the sum in 14) reduces to π M22m 1 sgn π q i<j ijmached in π i j i q Γ q i + j 1) = Pf q i j i q Γ q i + j 1) ) 2 ij 2m 1 = Pf q i+2 j i q Γ q i + j + ) ) 0 ij 2m = ε m 1 q mm 1)4m 5) m m 1 2k 1 q! Γ q 2k) Comparing with 1) we see that ε m = ε m 1 = = ε 1 = 1 Remark 5 Except the trivial but crucial point that ε m is independent of a b and q the above proof is a q-adaptation of Ciucu and Krattenthaler s proof 6 for the q 1 case of 12): 15) Pfj i)γa + i + j)) 0 ij 2m 1 = m 2k 1)!Γa + 2k 1) As we have shown that ε m is actually independent of q we could also reduce the proof directly to 15) by taking the limit q 1 in 12) 4 Link to Gram determinants of Askey-Wilson polynomials In this section we show that the determinant formula in Theorem 1 is actually related to a Gram determinant for the Askey-Wilson orthogonal polynomials see 20) This permits to enlighten the origin of the peculiar matrix coefficients 2) Let {p n x)} be a sequence of orthogonal polynomials with respect to a measure dµ and {φ k } and {ψ k } be two sequences of polynomials such that φ k and ψ k are

8 8 V J W GUO M ISHIKAWA H TAGAWA AND J ZENG of exact degree k Then µ 00 µ 01 µ 0n µ 10 µ 11 µ 1n 41) = C p n x) µ 0 µ 1 µ n φ 0 x) φ 1 x) φ n x) where µ ij = ψ i x)φ j x)dµ and C represents a factor of normalization Note that the three variables x θ z are related each other as follows: z = e Iθ x = cos θ x = z + z 1 2 For a b c d < 1 and n 0 let h n := h n a b c d q) with h 0 = abcd; q) q ab ac ad bc bd cd; q) h n = h 0 1 q abcd)q ab ac ad bc bd cd; q) n 1 q 2 abcd)abcd; q) n Then the orthogonality of Askey-Wilson polynomials reads p m cos θ; a b c d; q)p n cos θ; a b c d; q)wcos θ) dz 4πiz = h n 42) δ mn h 0 C where the contour C is the unit circle with suitable deformations see 4) and the weight function wx) := wx a b c d; q) is defined by wcos θ) = e 2Iθ e 2Iθ ; q) h 0 ae Iθ ae Iθ be Iθ be Iθ ce Iθ ce Iθ de Iθ de Iθ ; q) Except the constant factor C the following representation for the Askey-Wilson polynomials in terms of moments was already given by Atakishiyev and Suslov 2 Their proof was based on a generalization of Hahn s approach and Wilson 20 suggested a method to evaluate this determinant directly Theorem 41 For n 1 0 i n 1 and 0 j n let 4) ab; q) i+j A ij = ac ad; q) i bc bd; q) j abcd; q) i+j and A nj = bz b/z; q) j Then 44) deta ij ) 0 ij n = C p n x; a b c d; q) where x = z + z 1 )/2 and C = 1) n a n)/2 b nn+1)/2 q n)2)/6 i=0 ab ac ad bc bd cd q; q) i abcd; q) n+i

9 A QUADRATIC FORMULA FOR BASIC HYPERGEOMETRIC SERIES 9 Proof By 42) we have dz wcos θ) C 4πiz = 1 It follows that dz wx a b c d; q)az a/z; q) i bz b/z; q) j C 4πiz = h 0aq i bq j c d q) 45) wx aq i bq j c d; q) dz h 0 a b c d q) 4πiz = ac ad; q) i bc bd; q) j ab; q) i+j abcd; q) i+j C which coincides with A ij in 4) for i j = 0 n Thus by choosing ψ i = az a/z; q) i and φ j = bz b/z; q) j in 41) we obtain the determinant in 44) It remains to compute the factor C As p n x; a b c d; q) = 2 n abcdq ; q) n x n + lower terms bz b/z; q) n = 1 2bxq k + b 2 q 2k ) comparing the coefficients of x n in 41) we get 46) C = 1)n b n q n)/2 abcdq ; q) n deta ij ) 0 ij The determinant in 46) is essentially the Hankel determinant associated to the moments of little q-jacobi polynomials see 10) ) ab; q)i+j det = ab) n) 2 q n)n 2) q ab cd; q) k 47) abcd; q) i+j abcd; q) k+ 0 ij which is also a special case of 8) It follows from 47) that ) ab; q) i+j 48) det ac ad; q) i bc bd; q) j abcd; q) i+j = ab) n) 2 q n)n 2) j=1 0 ij ac ad bc bd; q) j Substituting this in 46) we recover the factor C in formula 44) q ab cd; q) k abcd; q) k+ Remark 42 Using 48) formula 44) can also be proven from the Desnanot-Jacobi adjoint matrix theorem and a special case of the known contiguous relation see 1 )) 49) rφ s aq A bq B ; q z r φ s a A b B ; q z = 1)1+s r za b) 1 b)1 bq) r 1 i=1 1 A i) s 1 i=1 1 B i) r φ s where A = A 1 A r 1 ) and B = B 1 B s 1 ) for r s 2 aq Aq bq 2 Bq ; q q1+s r z Actually it is not hard to see that Theorems 1 and 41 are equivalent Proposition 4 The two determinant formulae 44) are ) equivalent

10 10 V J W GUO M ISHIKAWA H TAGAWA AND J ZENG Proof Let z = e Iθ and x = cos θ Consider the matrix A := A ij ) 0 ij n where A ij are given in 4) Upon multiplying the j 1)st column of A by 1 2bxq j 1 + b 2 q 2j 2 and subtracting from the jth column for j = n n 1 1 the last row of the matrix becomes 1 0 0) and A ij 1 2bxq j 1 + b 2 q 2j 2 )A ij 1 = ac ad; q) i bc bd; q) j 1 B ij where B ij is given in 2) for 0 i n 1 and 1 j n Hence the two formulae 44) and ) are equivalent By 42) the linear functional L : Cx C associated to the orthogonal measure of the Askey-Wilson polynomials has the explicit integral expression: L x n) ) z + z 1 n 410) = wcos θ) dz 2 4πiz It follows from 45) with j = 0 and i = n that 411) C L az a/z; q) n ) = ab ac ad; q) n abcd; q) n Clearly if we take ψ i x) = φ i x) = x i for i 0 in 41) then µ ij = L x i+j) are the moments of Askey-Wilson polynomials and we obtain another determinant expression for the Askey-Wilson polynomials Such a formula would be interesting if we have a simple formula for the moments 410) Recently Corteel et al 4 and Ismail-Rahman 5 have published a double-sum formula for the moments of Askey-Wilson polynomials We would like to point out that their formula does follow straightforwardly from 411) and the Newton interpolation formula see 17 Chapter 1) that we recall below Theorem 44 Newton s interpolation formula) Let b 0 b 1 b be distinct complex numbers Then for any polynomial f of degree less than or equal to n we have n k ) fb j ) 412) fx) = k r=0r j b x b 0 ) x b k 1 ) j b r ) j=0 Indeed if b j = q j /a + aq j )/2 for j = 0 n 1 then j 1 b j b r ) = 1) j 2 j a j q j 2) q q 2j+1 /a 2 ; q) j r=0 k r=j+1 b j b l ) = 1) k j 2 j k a j k q j+1)k j) k j 2 ) q a 2 q 2j+1 ; q) k j As x = z + 1/z)/2 we have x b 0 ) x b k 1 ) = 1) k 2 k a k q k 2) az a/z; q)k Substituting the above into 412) we obtain Proposition 1 of 4 ie n k q k j2 a 2j f q j a+q j /a) 2 41) fx) = az a/z; q) k q q 2j+1 /a 2 ; q) j q q 2j+1 a 2 ; q) k j j=0

11 A QUADRATIC FORMULA FOR BASIC HYPERGEOMETRIC SERIES 11 In particular applying the linear functional L to the two sides of 41) with fx) = t + x) n we obtain Theorem 11 of 4 with a typo corrected which also appears in 9 Theorem 45 Corteel-Stanley-Stanton-Williams Ismail-Rahman) For any fixed t C we have n Lt + x) n ac ab ad; q) k k q k j2 a 2j t + qj a+q j /a) n 2 414) ) = abcd; q) k q q 2j+1 /a 2 ; q) j q q 2j+1 a 2 ; q) k j j=0 Remark 46 The proof of the above formula in 4 9 was based on a result of Ismail and Stanton 4 Theorem which is equivalent to 41) Since Ismail and Stanton s formula was originally proved using the Askey-Wilson operator in 5 Theorem 20 our proof seems more accessible for people who are not familiar with Askey-Wilson operator The t = 0 of 414) is the starting point of 12 Finally it is interesting to note that the following special case of Newton s formula 412) has been rediscovered recently by Mansour et al 15 Corollary 47 We have 415) where x + a 0 ) x + a ) = un k) = k r=0 n un k)x b 0 ) x b k 1 ) j=0 b r + a j ) k j=0j r b r b j ) for k = 0 n Remark 48 Clearly the connection coefficients un k) in 415) are characterized by the recurrence un k) = un 1 k 1) + a + b k )un 1 k) with boundary conditions un 0) = i=0 a i+b 0 ) and u0 k) = δ 0k the Kronecker delta function) Hence we recover the main result of Mansour et al 15 see also 19) Acknowledgements This paper was presented at the 25th International Conference on Formal Power Series and Algebraic Combinatorics FPSAC 1) Paris France June We are grateful to an anonymous referee for the conference FPSAC 1 for pointing out the link to Gram determinants of Askey-Wilson polynomials and to Sergei Suslov for bringing the reference 2 to our attention References 1 M Aigner A Course in Enumeration Graduate Texts in Mathematics Vol 28 Springer N M Atakishiyev and S K Suslov On the Askey Wilson polynomials Constr Approx ) pp 6 69 R Askey and J Wilson Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials Mem Amer Math Soc ) no 19 4 S Corteel R Stanley D Stanton and L Williams Formulae for Askey-Wilson moments and enumeration of staircase tableaux Trans Amer Math Soc ) no

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