Some Combinatorial and Analytical Identities

Transcription

1 Some Combinatorial and Analytical Identities Mourad EH Ismail Dennis Stanton November 6, 200 Abstract We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu We use the theory of basic hypergeometric functions, and generalize these identities We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which are not in the hierarchy of basic hypergeometric series We demonstrate that a Lagrange interpolation formula always leads to verywell-poised basic hypergeometric series As applications we prove that the Watson transformation of a balanced 4φ 3 to a very-well-poised 8φ 7 is euivalent to the Rodrigues-type formula for the Askey-Wilson polynomials By applying the Leibniz formula for the Askey-Wilson operator we also establish the 8φ 7 summation theorem Filename: FuLascoux-05 Reference: Annals of Combinatorics, 20 AMS Subect Classification 200 Primary: 05A9 and 33D5, Secondary: 05A30 and 33D70 Key words and phrases: partitions, identities of Chen and Liu, Dilcher, Fu and Lascoux, Prodinger and Uchimura, Summation theorems, polynomial expansions, bibasic sums, and Watson transformation, the Gasper identity, Lagrange type interpolation Introduction The interest in combinatorial identities goes back a long way but the interest in the combinatorial -identities is of a more recent vintage Riordan s book [26] crystalized the interest in combinatorial identities but it appeared before the combinatorial community realized the importance of -series outside the theory of partitions Since the 970 s it was realized that combinatorial identities and special function identities enrich and complement each other Some of the early combinatorial proofs of - series identities are in [2] and [5] This work two main goals Our first is to give new proofs and generalizations of the identities in 2, and to identify their origin within the theory of basic hypergeometric functions For example we show that not only the left and right sides of (2) are eual, but also that each sum can be found explicitly (Theorem 2) We show that (23) follows from the 2 φ to 3 φ transformation, [4, (III8)], and we generalize (22), (24), (28) in Theorems 22, 23, and 24 That such basic hypergeometric proofs exist is not surprising, moreover they indicate the depth of the identity Our second goal is to shed some new light on -series by showing how they follow from polynomial expansions We obtain new identities which do not follow from the basic hypergeometric framework The Watson transformation expresses a terminating very-well-poised 8 φ 7 as a terminating balanced 4 φ 3 Research supported by a grant from King Saud University in Riyadh and by Research Grants Council of Hong Kong under contract # 040

2 In 5 we show that the Watson transformation is simply the Rodrigues type formula for the Askey- Wilson polynomials, a surprising result Similarly the Whipple transformation is the Rodrigues type formula for the Wilson polynomials We follow the standard notation for -series as in [5] and [4] 2 The combinatorial identities In this section we prove (2), (23), and (25) and generalize (22), (24), (28) These are the combinatorial identities of Chen and Liu, Dilcher, Fu and Lascoux, Prodinger, and Uchimura We give uick proofs of these results and and state the generalizations in Theorems 22, 23, and 24 Theorem 2 sharpens (2) Fu and Lascoux [] used the Newton and Lagrange interpolation to prove several combinatorial identities One such identity is the following identity of Uchimura [27] (2) [ ] n ( ) (+ 2 ) +m / [m ] + This generalized the case m 0 which was proved in [28] Fu and Lascoux also treated the Dilcher identity [0] (22) [ ] n ( ) ( 2)+m ( ) m m n In another paper Fu and Lascoux [2] proved the identities (23) (z; ) n+ (; ) n 0 [ ] n ( ) x ( /x; ) z m m ( ) (z; ) x, (; ) 0 (24) [ ] n ( x) ( /x; ) ( ) m m ( ) [x ( ) ] 2 m n P m k2 k m k2 ( k ) (25) Prodinger [24] established the following -analogue of an earlier result of Kirchenhofer [23] 0, M [ ] n ( ) (+ 2 ) M ( )M (M+ 2 ) [ ] n M 0, M We show that (25) is an immediate conseuence of our version of (2) M M We establish (2) by establishing a stronger statement, that each side is summable Theorem 2 Each side of (2) is eual to m (; ) n ( m ; ) n+ 2

3 Proof A special case of the terminating 2 φ evaluation is [4, (II7)] [ ] n ( ) (+ 2 ) z z 2 φ n, z z ; n+ (; ) n (26) (z; ) n+ 0 Put z m to evaluate the left side of (2) One can easily verify that the left side L of (2) is the partial fraction expansion in z m for the right side R It also follows from a 3 φ 2 transformation [4, (III2)] R z lim ɛ 0 3 φ n, +ɛ, 2 + n+ɛ, z 2, n z n lim ɛ 0 3 φ n, n, 2 + n+ɛ, n /z, ɛ /z [ ( ) z n n, n 2φ n+ /z, /z (; ) n( n ] ; ) n (; ) n ( n n z n /z; ) n The 2 φ can be evaluated as a product, again by [4, (II7)], and the proof is complete Our version (26) of (2) immediately proves (25) Proof of (25) Fix an integer M between 0 and n, and write (26) as [ ] [ n ( ) (+ 2 ) z (; ) n z M (z; ) M (z M+ ( )M ( ] M+ 2 ) ; ) n M (; ) M (; ) n M 0, M If f(z) {(z; ) M (z M+ ; ) n M }, then the uantity in the suare bracket is f(z) f( M ) Taking the limit as z M of the resulting identity yields (25) Proof of (23) Recall the basic hypergeometric transformation [4, (III8)] 2φ n, B C, Z (C/B; ) ( n B n 3φ n, B, /Z (C; ) n B n /C, Z ) C The right side of (23) is a limit of a 2 φ Upon applying the above transformation lim ɛ 0 2 φ n, z + ɛ n, x ( n /z; ) n ( n z n 3φ n, z, /x ; ) n z, n+ x, we obtain the left side of (23) For the proof of (22) we prove a more general identity Theorem 22 For any positive integer m, [ ] n ( ) ( 2)+m ( z ) m+ (; ) n (z; ) n nm ( z) 22 ( z 2 ) 2 nn ( z n ) n It is evident that (22) is the special case z of Theorem 22 Proof Rewrite (26) as [ ] n ( ) ( 2) ( ) z (; ) n (z; ) n Now differentiate the above identity m times with respect to z to obtain Theorem 22 3

4 The final variation of (26) generalizes (24) Theorem 23 For any positive integer m (; ) n n [ ] n ( x) ( /x; ) ( z ) m m ( ) [x ( ) ] (; ) (z ; ) n+ 2 m n Proof Observe that the Chu-Vandermonde sum [4, (II6)] implies (27) 0 ( (z; ) lim (; ) ɛ 0 2 φ n, z + n ɛ ), P m k2 k m k2 ( z k ) ( n /z; ) n ( n ; ) n z n (z; ) n (; ) n Now subtract the right-hand side of (27) from the left-hand side of (23) and the left-hand side of (27) from the right-hand side of (23) to establish (z; ) n+ (; ) n [ ] n ( x) ( /x; ) z ( ) (z; ) (x ( ) ) (; ) Dividing the above identity by (z; ) n+ /(; ) n and differentiating m times with respect to z proves Theorem 23 In a recent paper Chen and Liu [7] generalized earlier work of Alladi [] involving weighted partition theorems Chen and Liu were interested in Franklin type involutions, and as an application of their techniue, they proved the identity (28) 2mn ( 2mn+2m ; 2m ) (a 2mn+ ; 2 ) n0 + ( a) k k2 k k [ (m )] The case a is due to Andrews, see [3, p 57] We generalize (28) in Theorem 24, and give its combinatorial interpretation in Theorem 25 We need only the -binomial theorem Theorem 24 For any positive integer m, n0 + 2mn ( 2mn+2m ; 2m ) (a 2mn+ ; 2 ) (ab 2mn+ ; 2 ) (ab) k (/b; 2 ) k k k It is clear that (28) is the special case b 0 of Theorem 24 Proof Using the -binomial theorem n0 (a; ) n (; ) n z n (az; ) (z; ), z <, [ (m )] 4

5 we write the left side of Theorem 24 as 2mn ( 2mn+2m ; 2m ) n0 and the proof is complete ( 2m ; 2m ) (/b; 2 ) k ( 2 ; 2 ) k (ab) k (/b; 2 ) k ( 2 ; 2 ) k (ab) k ( 2m ; 2m ) ( 2m+2mk ; 2m ) (/b; 2 ) k ( 2 ; 2 ) k (ab) k 2mnk n0 2mn(k+) ( 2m ; 2m ) n (/b; 2 ) k (ab) k (2m ; 2m ) k ( 2 ; 2, ) k Next we give an integer partition interpretation of Theorem 24 which generalizes Theorem 72 in [7] We give two sets of integer partitions whose generating functions are the respective sides of Theorem 24 For the left side, let A k,s,m (n) be the set of integer partitions λ of n such that the even parts of λ are distinct non-negative integers, 2 each even part of λ is a multiple of 2m, 3 the smallest part of λ is even, 4 the odd parts of λ form an overpartition with k total parts and k s barred parts An example is λ (3, 3, 2, 9, 7, 5, 5, 5, 4) A 7,5,2 (73) We see that if 2m is the smallest part of λ, the generating function of A k,s,m (n) is k,s,n 0 A k,s,m (n) a k b s n 0 2m ( 2m+2m ; 2m ) ( a 2m+ ; 2 ) (ab 2m+ ; 2 ), which is nearly the left side of Theorem 24, with a a, b b If the even parts greater than 2m are weighted by, we do obtain the left side of Theorem 24 The right side of Theorem 24, with a a, b b is + k a k k p (b + 2p ) k [ (m )] For the combinatorial interpretation of the coefficient of a k b s, we choose s parts of size, and k s distinct odd parts, 2k θ > θ 2 > > θ k s each at most 2k Then µ (θ (2(k s) ), θ 2 (2(k s) 3),, θ k s ) is a partition with even parts lying inside a (k s) 2s rectangle The s s, and the k s odd parts can be concatenated to obtain a partition γ with k parts, γ (2(k s), 2(k s) 3,,,,, ) The product k [ (m )] is the generating function for partitions γ 2 with exactly k parts, 0 s allowed, each part even, with difference of consecutive parts at most 2(m ) Define µ 2 γ + γ 2, the partition obtained by adding the respective parts We can now define a set B k,s,m (n) for the right side of Theorem 24 Let B k,s,m (n) be the set of pairs of integer partitions (µ, µ 2 ), µ + µ 2 n, such that µ is a partition with even parts which lies inside a (k s) 2s rectangle, 2 µ 2 (m,, m k ) has exactly k parts, all of which are odd, the first k s of which are distinct, 5

6 3 m i m i+ 2m for i k s, 4 m i m i+ 2(m ) for k s i k, 5 m k 2m The right side of Theorem 24 is k,s,n 0 B k,s,m (n) a k b s n Theorem 72 in [7] is the s 0 case of the next Theorem Theorem 25 For any positive integer m, B k,s,m (n) ( ) (#even parts of λ) λ A k,s,m (n) For example if n 0, k 4, s 3, and m 2, B 4,3,2 (0) {(, 53), (, 333), (2, 33), (4, 3), (6, )}, A 4,3,2 (0) {70, 70, 530, 530, 530, 3330, 3330, 430, 430} 3 Polynomial Expansions In the previous section we applied the beginning results in basic hypergeometric series to derive recent combinatorial identities In this section we will derive new identities, which are not special cases of the basic hypergeometric literature We use polynomial expansions which are extensions of Taylor s theorem Along the way we give very short proofs of Jackson s summation theorem and Watson s transformation for very well-poised 8 φ 7 s (3) Let x (z + /z)/2, z e iθ, x cos θ The Askey-Wilson basis for the vector space of polynomials in x is (32) n φ n (x; a) (ae iθ, ae iθ ; ) n ( 2ax i + a 2 2i ), n 0,, i0 The uestion of finding the coefficients when an arbitrary polynomial is expanded in this basis is answered using the Askey-Wilson operators [8], which generalize the derivative Put The Askey-Wilson operator D is defined by f(x) f((z + /z)/2) f(z) It is easy to see that (33) (34) (D f)(x) f( /2 z) f( /2 z) ( /2 /2 )[(z /z)/2] D φ n (x; a) 2a( n ) φ n (x; a /2 ) D φ n (x; a) 2a( n ) φ n+ (x; a /2 ) The expansions we shall use are embodied in the following theorem 6

7 Theorem 3 Let and assume that f(x) is a polynomial Then with x n [a n/2 + n/2 /a]/2, 0 < <, 0 < a <, f k,φ f(x) Theorem 3 for the φ n basis is in [7] f k,φ φ k (x; a), ( )k (2a) k (; ) k k(k )/4 (D k f)(x k ) Theorem 32 The action of D n is given by 2n n( n)/4 D n f(x) ( /2 /2 ) n [ ] n k(n k) z 2k n f( (n 2k)/2 z) k ( +n 2k z 2 ; ) k ( 2k n+ z 2, ; ) n k where x cos θ, z e iθ Theorem 32 is due to S Cooper [8] It is clear that Theorem 32 is a summation result whenever D n f(x) is explicitly known For example, from (33) we see that f(x) φ m (x; A), D n f(x) ( m n+ ; ) n 2( n 2) (2a)n ( ) n φ m n(x; A n/2 ) In this case Theorem 32 becomes the terminating very well-poised 6 φ 5 evaluation [4, II20] By taking a product of two φ k s we obtain a terminating very-well poised 8 φ 7 from Theorem 32 It only remains to choose an appropriate product for a summation result Proposition 33 For any positive integer, φn+ (x; A) (2B)n 2( n 2) D n φ (x; B) ( ) n φ (x; A n/2 ) (; ) n+ (AB ; ) φ n+ (x; B n/2 n (A/B; ) n ) (; ) Proof The Leibniz rule [7, (22)] for the Askey-Wilson operator is (35) where η k is the map D n (f(x)g(x)) [ ] n k(k n)/2 (η k (D n k f))(η k n (D k g)), k (η k f)(x) f( k/2 z) When (35), and (33)- (34) are applied in Proposition 33, the sum which results is a -balanced 3 φ 2, which is evaluable [4, II2] 7

8 Observe the unusual fact that the right-hand side of the formula in Proposition 33 is a constant multiple of D n φ n+ (x; A) D n, φ (x; B) a very curious fact Applying Proposition 33 to Theorem 32 does yield Jackson s terminating very well-poised 8 φ 7 evaluation [4, II22], the parameters are a z 2 n, b Az n/2+, c z n/2+ /A, d Bz n/2, and e z n/2 + /B If we choose f(x) to be a product of two arbitrary φ s, f(x) φ m (x; A)φ (x; B), then the right side of Theorem 32 becomes a very-well poised 8 φ 7 The Leibniz rule (35) yields a -balanced 4 φ 3 for the left side of Theorem 32 The resulting euality is Watson s transformation [4, (III8)] If a general product of p + φ s is chosen, the resulting euality is Andrews transformation [4, Theorem 4] of a very-well poised 2p+6 φ 2p+5 to a p-fold sum We combine Theorems 3 and 32 in the next result Theorem 34 will be applied throughout the rest of this section Theorem 34 For polynomials f of degree at most n and with x cos θ, we have the expansion (, a 2 ; ) n (ae iθ, ae iθ ; ) n f(x) a 2 2k a 2 ( n, a 2, ae iθ, ae iθ ; ) k (, a 2 n+, ae iθ, ae iθ ; ) k k(+n) f(a k ) Proof Combine Theorem 3 and Theorem 32 to find that f(x) 0 k n (k )2 +k f(a k )(ae iθ, ae iθ ; ) k a 2k 2 (, a 2 +2k 2 ; ) (, 2 2k+ /a 2 ; ) k, holds for polynomial f Replace k by k + and the new sum becomes lim ɛ 0 3 φ k n, a k e iθ, a k e iθ 2 + k n+ɛ, a 2 2k+ ɛ, (ak+ e iθ, a k+ e iθ ; ) n k (, a 2 2k+, ; ) n k by the -Pfaff-Saalschütz sum, [4, (II2)] Theorem 34 now follows from simple manipulations Theorem 34 can be considered as another polynomial expansion Upon multiplying both sides by φ n (x; a) note that φ k (x; a)φ n (x; a) φ k (x; a) φ n+ (x; a) 2xa k + a 2 2k so that the k th term of the right side is a polynomial in x Theorems 3 and 34 lead to distinct results Theorem 34 is related to very-well-poised sums, while Theorem 3 is related to balanced sums From the classical theory of interpolation, Theorem 3 is an expansion using divided difference operators while Theorem 34 is the Lagrange interpolation, [9], [6] The first application of Theorem 34 chooses f(x) (2x + b) n Corollary 35 For non-negative integers n we have a n (, a 2 ; ) n (ae iθ, ae iθ ; ) n (2x + b) n a 2 2k a 2 ( n, a 2, ae iθ, ae iθ ; ) k (, a 2 n+, ae iθ, ae iθ ; ) k k [ + ab k + a 2 2k ] n 8

9 By replacing n by n + s in Corollary 35 then euating coefficients of b s we establish the identity (36) n+s (, a 2 ; ) n+s (ae iθ, ae iθ ; ) n+s (2a cos θ) n a 2 2k a 2 ( n s, a 2, ae iθ, ae iθ ; ) k (, a 2 n+s+, ae iθ, ae iθ ; ) k k(+s) [ + a 2 2k ] n The special case b 0 of Corollary 35, or euivalently the case s 0 of (36), is the attractive identity (37) (a 2, ac, a/c, n ; ) k a 2 2k (, a/c, ac, a 2 n+ ; ) k a 2 k ( + a 2 2k ) n (a2, ; ) n a n (c + /c) n (a/c, ac; ) n The above identity is a partial fraction expansion in c and can be proved by computing residues The limiting case s of (36) is the sum (38) (2a cos θ) n (, a 2 ; ) (ae iθ, ae iθ ; ) a 2 2k a 2 (a 2, ae iθ, ae iθ ; ) k (, ae iθ, ae iθ ; ) k ( ) k nk+(k+ 2 ) [ + a 2 2k ] n Our next result is a bibasic sum Corollary 36 For non-negative integers n we have (, a 2 ; ) n (ae iθ, ae iθ ; ) n (be iθ, be iθ ; p) n a 2 2k a 2 ( n, a 2, ae iθ, ae iθ ; ) k (, a 2 n+, ae iθ, ae iθ ; ) k k(+n) (ab k, b k /a; p) n It must be noted that the known bibasic results are proved using telescopy [4] but Corollary 36 does not seem to be suitable for a proof by telescopy The limiting case n is our earlier result, [20, (53)], (39) (, a 2 ; ) (ae iθ, ae iθ ; ) (be iθ, be iθ ; p) a 2 2k a 2 (a 2, ae iθ, ae iθ ; ) k (, ae iθ, ae iθ ; ) k ( ) k (k+ 2 ) (ab k, b k /a; p), which is valid for 0 < p < <, or 0 < p < and b < a The case p of Corollary 36 is Corollary 35 The following examples use the very-well-poised functions W The W notation, due to W N Bailey, is defined by (30) 3+mW 2+m (a; b,, b m ;, z) 3+mφ 2+m ( a, a, a, b,, b m a, a, a/b,, a/b m ) ; z The very special case p of Corollary 36 is the following corollary 9

10 Corollary 37 The summation theorem holds (, a 2 ; ) n (ae iθ, ae iθ ; ) n (be iθ, be iθ ; ) n (b/a, ab; ) n 8 W 7 (a 2, ae iθ, ae iθ, ab n, a/b, n ;, ), Corollary 37 is a special case of Jackson s -analogue of Dougall s 7 F 6 sum [4, (II22)] The general theorem of Jackson contains one more parameter This is surprising because if we calculate the coefficients f k,φ for f(cos θ) (be iθ, be iθ ; ) n then apply the expansion as in Theorem 3 we will get the -analogue of the Pfaff-Saalschütz theorem, as noted in [7] This indicates that the Pfaff-Saalschütz theorem is euivalent to Corollary 37 Corollary 38 We have the following 0 W 9 summation theorem (, a 2 ; ) n (ae iθ, ae iθ ; ) n (be iθ, be iθ ; ) s (ce iθ, ce iθ ; ) n s (b/a, ab; ) s (ac, c/a; ) n s 0 W 9 (a 2, ae iθ, ae iθ, ab s, ac n s, a/b, a/c, n ;, ) The proof consists of taking f(cos θ) (be iθ, be iθ ; ) s (ce iθ, ce iθ ; ) n s in Theorem 34 Similarly we have the following theorem Corollary 39 For s + s s m n we have the summation theorem (, a 2 ; ) n m (b e iθ, b e iθ ; ) s (ae iθ, ae iθ ; ) n (ab, b /a; ) s 6+2m W 5+2m (a 2, ae iθ, ae iθ, n, a/b, ab s,, a/b m, ab m sm ;, ) Corollary 39 gives the sum of a terminating W function To find a nonterminating version we first replace m by m +, then let n, s m+ + while N n s m+ remains constant Let k be the summation index in the W series The terms involving n or s m+ are ( n, ab m+ sm+ ; ) k (a 2 n+, a sm+ /b m+ ; ) k k(+n) as n Thus a limiting case of Corollary 39 is (3) (, a 2, b m+ e iθ, b m+ e iθ ; ) (ae iθ, ae iθ, ab m+, b m+ /a; ) m ( bm+ a ) k, (b e iθ, b e iθ ; ) s (ab, b /a; ) s 6+2m W 5+2m (a 2, ae iθ, ae iθ, a/b m+, a/b, ab s,, a/b m, ab m sm ;, N b m+ a ), where N s + + s m and b m+ < a N if the W series does not terminate The summation theorem (3) is due to George Gasper [3] In fact Corollary 39 is euivalent to (3) because we may choose b m+ a N+ in (3) and recover Theorem 39 Note that (38) is the limiting case b m+ 0 of (3) when b, m n, s s 2 s m An interesting limiting case of Corollary 39 is to let z e iθ + The result is (, a 2 ; ) n m b s s(s )/2 (32) a n n(n+)/2 (ab, b /a; ) s 4+2m W 3+2m (a 2, n, a/b, ab s,, a/b m, ab m sm ;, ) For another instance where a basic hypergeometric series is evaluated at, see [4, 9] We now consider a bivariate version of Theorem 34 0

11 Theorem 30 Let f(x, y) be a polynomial of degree at most n in x and in y With x cos θ, y cos φ, the following expansion holds (,, a 2, b 2 ; ) n (ae iθ, ae iθ, be iφ, be iφ ; ) n f(x, y), ( a 2 2 )( b 2 2k ) ( a 2 )( b 2 ) (+k)(+n) ( n, a 2, ae iθ, ae iθ ; ) ( n, b 2, be iφ, be iφ ; ) k f(a (, a 2 n+, ae iθ, ae iθ ; ) (, b 2 n+, be iφ, be iφ, b k ) ; ) k Applying Theorem 30 to f(x, y) (αe i(θ+φ), αe i(φ θ), αe i(θ φ), αe i(θ+φ) ; ) n, a polynomial in x and y of degree 2n, leads to the next result Corollary 3 The following summation theorem holds (,, a 2, b 2 ; ) 2n (αe i(θ+φ), αe i(φ θ), αe i(θ φ), αe i(θ+φ) ; ) n (αab, α/ab; ) n (ae iθ, ae iθ, be iφ, be iφ ; ) 2n 2, (αab n, ab/α; ) +k a 2 2 ( 2n, a 2, ae iθ, ae iθ ; ) (αab, ab n /α; ) +k a 2 (, a 2 2n+, ae iθ, ae iθ ; ) b2 2k b 2 ( 2n, b 2, be iφ, be iφ ; ) k (, b 2 2n+, be iφ, be iφ ; ) k (+k)(n+) (αb/a; ) n+k (αa/b; ) n+ k (αa k /b; ) k (αb k /a; ) (αb/a; ) k (αa/b; ) A bivariate version of Corollary 35 is (33) a n (, a 2 ; ) n (ae iθ, ae iθ ; ) n, b n (, b 2 ; ) n (be iφ, be iφ ; ) n (2cosθ + 2 cos φ + c) n b 2 2 b 2 a 2 2k a 2 ( n, b 2, be iφ, be iφ ; ) (, b 2 n+, be iφ, be iφ ; ) +k ( n, a 2, ae iθ, ae iθ ; ) k (, a 2 n+, ae iθ, ae iθ ; ) k [a k + b + abc k+ + ab 2 2+k + a 2 b 2k+ ] n Again replace n by n + s then euate the coefficient of c s The result is (34) a n (, a 2 ; ) n+s b n (, b 2 ; ) n+s (ae iθ, ae iθ ; ) n+s (be iφ, be iφ (2cosθ + 2 cos φ) n ; ) n+s n+s, b 2 2 b 2 a 2 2k a 2 ( n s, b 2, be iφ, be iφ ; ) (, b 2 n+s+, be iφ, be iφ ; ) (+k)(+s) ( n s, a 2, ae iθ, ae iθ ; ) k (, a 2 n+s+, ae iθ, ae iθ ; ) k (a k + b ) n ( + ab +k ) n 4 The Wilson basis All of the results of 3 can be given for the corresponding Wilson basis w n (x; a) (a + i x, a i x) n,

12 and Wilson operator W (Wf)(x) f(z + i/2) f(z i/2) 2i, x where f(x) f(z), z x, The analogues of Theorems 3, 32, and 34 are given here Theorem 42 is due to Cooper Theorem 4 Let and assume that f(x) is a polynomial Then f(x) y k (a + k/2) 2, f k,w w k (x; a), f k,w k! (Wk f)(y k ) Theorem 42 The action of W n is given by W n f(x) n ( ) k f(z + i(n 2k)/2)), z x, k ( 2iz + + n 2k) k (2iz + + 2k n) n k Theorem 43 For polynomials f of degree at most n we have the following expansion a + k a n!(2a + ) n (a + + i x) n (a + i x) n f(x) ( n) k (2a) k (a + i x) k (a i x) k k!(2a + n + ) k (a + + i x) k (a + i x) k f(i(a + k)) It is clear that Theorem 43 is closely related to very-well-poised hypergeometric series 5 The Watson Transformation In this section we reverse the use of Theorem 32, namely use it to write very-well-poised series as the n th iterate of an Askey-Wilson operator When this is realized for the Askey-Wilson polynomials, we see that the Rodrigues formula is euivalent to Watson s transformation The Askey-Wilson polynomials have the basic hypergeometric representation p n (x; t ) t n (t t 2, t t 3, t t 4 ; ) n 4φ n, t t 2 t 3 t 4 n, t e iθ, t e iθ (5) 3 t t 2, t t 3, t t 4,, where t stands for the ordered uadruple (t, t 2, t 3, t 4 ), [6], [8] Their weight function is (52) w(x, t ) (e 2iθ, e 2iθ ; ) 4 (t e iθ, t e iθ ; ) 2ie iθ (e 2iθ, e 2iθ ; ) x 2 4 (t, e iθ, t e iθ ; ) with x cos θ (, ) They have the Rodrigues type formula [6], [4], [8] (53) n n(n )/4 p n (x; t) 2 w(cos θ; t) Dn [w(x; n/2 t)] 2

13 From Theorem 32 it follows that the right-hand of (53), with z e iθ, is (54) k(+n k) z 2k n [ ] n z 2 n 2k k z 2 (/z 2 ; ) k (z 2 ; ) n k ] [ n k z 2 n 2k z 2 (n k)(+k) z n 2k (/z 2 ; ) n k (z 2 ; ) k After routine manipulations we arrive at the representation 4 (t z; ) n k (t /z; ) k 4 (t z; ) k (t /z; ) n k (55) p n (cos θ; t) zn 4 (t /z; ) n (/z 2 ; ) n 8 W 7 ( n z 2 ; n, t z, t 2 z, t 3 z, t 4 z;, 2 n /t t 2 t 3 t 4 ) The euality of the right sides of (55) and (5) is the Watson transformation [4, (III8)] after the application of the iterated Sears transformation [4, (II5)] Note that Ismail [7] proved the Sears transformation using -Taylor series in the Askey-Wilson operator This is reproduced in Chapter 2 of [8] Rodrigues formulas for other very-well-poised series, including 0 W 9 s, may be found in this manner, see [22] Acknowledgments We thank the referee for his/her careful reading of the manuscript and for helpful suggestions References [] K Alladi, A combinatorial study and comparison of partial theta identities of Andrews and Ramanuan, Ramanuan J, to appear [2] G E Andrews, On the foundations of combinatorial theory V : Eulerian differential operators, Studies in Applied Math 50 (97), [3] G E Andrews, The Theory of Partitions, Cambridge University Press, paperback edition, Cambridge, 998 [4] G E Andrews, Problems and prospects for basic hypergeometric series, in Theory and Application of Special Functions, R Askey, ed, Academic Press, 975, pp [5] G E Andrews, R A Askey, and R Roy, Special Functions, Cambridge University Press, Cambridge, 999 [6] R Askey and J Wilson, Some basic hypergeometric orthogonal polynomials that generalizes Jacobi polynomials, Memoirs Amer MathSoc Numebr 37, 985 [7] W Y C Chen and E Liu, A Franklin type involution for suares, Adv Appl Math, to appear [8] S Cooper, The Askey-Wilson operator and the 6 φ 5 summation formula, preprint, March 996 [9] P J Davis, Interpolation and Approximation, Dover Publications, New York, 975 [0] K Dilcher, Some -identities related to the divisor function, Discrete Math 45 (995), [] A M Fu and A Lascoux, -identities from Lagrange and Newton interpolation, Adv Appl Math 3 (2003),

14 [2] A M Fu and A Lascoux, -identities related to overpartitions and divisor functions, Electronic J Combinatorics 2 (2005), # R 38 [3] G Gasper, Elementary derivations of summation and transformation formulas for -series, in Special functions, -series and related topics, Fields Inst Commun, 4, M E H Ismail, D R Masson, and M Rahman, eds, American Mathematical Society, Providence, 997, pp [4] G Gasper and M Rahman, Basic Hypergeometric Series, second edition Cambridge University Press, Cambridge, 2004 [5] J Goldman and G C Rota, On the foundations of combinatorial theory IV : Finite dimensional vector spaces and Eulerian generating functions, Studies in Applied Math 49 (970), [6] A O Guelfond, Calcul Des Différence Finies, French Translation, Dunod, Paris, 963 [7] M E H Ismail, The Askey-Wilson operator and summation theorems, in Mathematical Analysis, Wavelets, and Signal Processing, M Ismail, M Z Nashed, A Zayed and A Ghaleb, eds, Contemporary Mathematics, volume 90, American Mathematical Society, Providence, 995, pp 7 78 [8] M E H Ismail, Classical and Quantum Orthogonal Polynomials in one variable, Cambridge University Press, paperback edition, Cambridge, 2009 [9] M E H Ismail and D Stanton, Applications of -Taylor theorems, J Comp Appl Math 53 (2003), [20] M E H Ismail and D Stanton, -Taylor theorems, polynomial expansions, and interpolation of entire functions, J Approximation Theory 23 (2003), [2] M E H Ismail and D Stanton, -Taylor series and interpolation, preprint, 200 [22] M E H Ismail and D Stanton, Rodrigues type formulas, asymptotics, and biorthogonality, preprint, 200 [23] P Kirchenhofer, A note on alternating sums, Electronic J Combin 3 (2) (996), R7 0 pages [24] H Prodinger, Some applications of the -Rice formula, Random Structures Algorithms 9 (200), [25] M Rahman, An integral representation of a 0 ϕ 9 and continuous bi-orthogonal 0 ϕ 9 rational functions, Canad J Math 38 (986), no 3, [26] J Riordan, Combinatorial Identities, Wiley, New york, 960 [27] K Uchimura, A generalization of identities for the divisor generating function, Utilitas Math 25 (984), [28] L Van Hamme, Advanced Problem 6407, Amer Math Month 89 (982), Mourad EH Ismail, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong and King Saud University, Riyadh, Saudi Arabia ismail@mathucfedu Dennis Stanton, University of Minnesota, Minneapolis, MN USA stanton@mathumnedu 4