POSITIVITY PRESERVING TRANSFORMATIONS FOR q-binomial COEFFICIENTS

Transcription

1 POSITIVITY PRESERVING TRANSFORMATIONS FOR q-binomial COEFFICIENTS ALEXANDER BERKOVICH AND S OLE WARNAAR Abstact Seveal new tansfomations fo q-binomial coefficients ae found, which have the special featue that the kenel is a polynomial with nonnegative coefficients By studying the goup-like popeties of these positivity peseving tansfomations, as well as thei connection with the Bailey lemma, many new summation and tansfomation fomulas fo basic hypegeometic seies ae found The new q-binomial tansfomations ae also applied to obtain multisum Roges Ramanujan identities, to find new epesentations fo the Roges Szegö polynomials, and to make some pogess on Bessoud s genealized Bowein conjectue Fo the oiginal Bowein conjectue we fomulate a efinement based on a new tiple sum epesentations of the Bowein polynomials j () Intoduction q-binomial tansfomations In the liteatue on q-seies one finds numeous tansfomations of the type q 4 (q; q) L () (q; q) (L ) q 4 L j (q; q) ( j) (L j) and () j () q 8 (q; q) L (q, q (+) ; q) (L ) (q; q) q 8 j ( j) L, (L j) whee j and L ae integes such that j L (mod ) (Thoughout this pape the notation a b (c) instead of a b (mod c) will be used in equations fo bevity) Hee L L (q; q) L fo a {0,, L} (q; q) a (q; q) L a a q a 0 othewise is a q-binomial coefficient, n (a; q) n ( aq j ) j0 000 Mathematics Subject Classification Pimay 33D5; Seconday 33C0, 05E05 Key wods and phases Bailey Lemma, base-changing tansfomations, basic hypegeometic seies, Bowein conjectue, q-binomial coefficients, Roges Ramanujan identities, Roges Szegö polynomials Fist autho suppoted in pat by NSF gant DMS Second autho suppoted by the Austalian Reseach Council

2 ALEXANDER BERKOVICH AND S OLE WARNAAR is a q-shifted factoial, and (a,, a k ; q) n k (a j ; q) n Impotant featues of () and () ae (i) the sum ove a q-binomial coefficient multipied by a simple facto again yields a q-binomial coefficient, (ii) only the lowe enties of the q-binomial coefficients and a simple exponential facto on the ight depend on j, (iii) they can eadily be iteated As an example of this last point let us conside the simple q-binomial identity (3) ( ) j q ) L (j δ L,0, L j j L which is a special case of the finite fom of Jacobi s tiple poduct identity 3, Ch 3, Example (see also (77)) Replacing L by, multiplying both sides by j q (q; q) L (q; q) L (q; q) and summing ove using () with L L, j j and, yields (4) ( ) j q +( j ) j L (q L+ ; q) L L j j L This bounded vesion of Eule s pentagonal numbe theoem 4 is of the same fom as (3) and we may epeat the above pocedue to find the well-known bounded analogue of the fist Roges Ramanujan identity, 58 (5) ( ) j q +( j ) j L L L (q L+ ; q) L q L j j L Some known q-binomial tansfomations simila to () and (), but in which the base of the q-binomial coefficient is changed fom q to q o q 3, ae given by q 4 (q; q) L (6) (q ; q ) (L ) q 4 L j, (q; q) ( j) (L j) q (7) and (8) j () j () j () q 8 (q (+) ; q) L (q; q) L (q, q + ; q ) (L ) (q; q) q 4 (q; q) (3L ) (q 3 ; q 3 ) (L ) (q; q) q 8 j ( j) q 4 j ( j) assuming once again that j L (mod ) All of the above tansfomations ae of the fom (9) q 8 γ f L, (q) q 8 γj ( j) j () L (L j) L (L j) L (L j) q k q 3, q,

3 POSITIVITY PRESERVING TRANSFORMATIONS 3 with f L, (q) a polynomial in q o q /, which fo 0 < L has both positive and negative coefficients The issue of positivity of coefficients in polynomial expessions of the type given by the left-hand sides of (3), (4) and (5) has ecently eceived consideable attention in elation to conjectues of Bowein 8 and Bessoud 9 Fo this eason it is impotant to find q-binomial tansfomations a là (9) with f L, (q) a polynomial with nonnegative coefficients We will efe to such tansfomations as positivity peseving Indeed, applying a positivity peseving tansfomation to an identity like (3) with on the ight a polynomial with nonnegative coefficients esults in a new identity which again has a polynomial with nonnegative coefficients on the ight Outline In the next section five new, positivity peseving q-binomial tansfomations plus two elated, ational tansfomations ae poved In ode to establish the positivity of one of ou esults we genealize nonnegativity theoems of Andews fo q-binomial coefficients and of Haiman fo pincipally specialized Schu functions Goup-like elations among ou q-binomial tansfomations and those listed in the intoduction ae investigated in Section 3 This will give ise to numeous new tansfomation fomulas fo balanced and almost balanced basic hypegeometic seies The inveses of the tansfomations fo q-binomial coefficients ae established in Section 4 Again this will lead to seveal elegant new summation fomulas The elation between ou q-binomial tansfomations and the Bailey lemma is the subject of Sections 5 and 6 The eade may indeed have ecognized () and () as special cases of the odinay Bailey lemma in its vesion due to Andews 6 and Paule 4, and (6) (8) as special cases of base-changing extensions of the Bailey lemma discoveed by Bessoud, Ismail and Stanton 0 In Section 5 we show that ou new tansfomations coespond to new types of base-changing Bailey lemmas In Section 6 this is exploited to yield some new (and old) tansfomations fo basic hypegeometic seies The Sections 7 and 8 deal with simple applications of the q-binomial tansfomations of section In Section 7 new single and multisum identities of the Roges Ramanujan identities ae poved and in Section 8 we obtain a emakable new epesentation of the Roges Szegö polynomials Finally, in Section 9, we use the positivity peseving natue of ou esults to make some pogess on Bessoud s genealized Bowein conjectue In the last section we also pove new tiple-sum epesentation fo the Bowein polynomials and use this to fomulate a new conjectue that implies the oiginal Bowein conjectue Positivity peseving q-binomial tansfomations The eason that none of the tansfomations of the pevious section peseves positivity is not a vey deep one Setting q in (9) yields ( ) ( ) L f L, (), ( j) (L j) j () which has the unique solution f L, () δ L, Hence the only polynomial solution to (9) that peseves positivity is the less-than-exciting f L, (q) δ L, fo k and

4 4 ALEXANDER BERKOVICH AND S OLE WARNAAR γ 0 To get aound this poblem we need to modify (9), and in the following we look fo polynomials f L, (q) with nonnegative coefficients that satisfy () q 4 γ f L, (q) q L 4 γj, k ( j) L j q k j () o small vaiations heeof (see (8) below) To see that fom a positivity point of view () is indeed moe pomising than (9), let us again set q Multiplying both sides by x j and summing ove j using the binomial theoem gives f L, ()(x + x ) (x + x ) L This is eadily solved to yield () f L, () L ( L a solution that may well have q-analogues fee of minus signs In the emainde of the pape we will make extensive use of basic hypegeometic seies, and befoe pesenting ou solutions to () we need to intoduce some futhe notation 30 Fist, φ s a,, a b,, b s ; q, z ), φ s (a,, a ; b,, b ; q, z) k0 (a,, a ; q) k ( ) k q ) s + (k z k (q, b,, b s ; q) k Hee it is assumed that the b i ae such that none of the factos in the denominato is zeo, q 0 if > s + and q < wheneve the φ s is nonteminating Moeove, if the seies does not teminate then s + with z < if s + If it does howeve teminate one can evese the ode of summation as discussed in 30, Execise 4 An + φ seies is called balanced if z q and a a + q b b, well-poised if qa a b a + b and vey-well-poised if it is well-poised and a a 3 a / q We will always abbeviate such vey-well-poised seies by + W (a ; a 4,, a + ; q, z) Wheneve one of the numeato paametes in a q-hypegeometic seies is q n we assume n to be a nonnegative intege (Hence, povided the base of the seies is q (o q /, q /3 etc), the seies will teminate) Afte these definitions we etun to () Fo k it is not had to see that thee ae no factoizable solutions (two non-factoizable o non-q-hypegeometic solutions ae given in Section 9), and all ou esults will involve a change of base Thee is of couse ample pecedent fo base-changing tansfomations see, eg,, 3, 0, 8, 9, 30, 5, 53, 56 Ou fist esult is of a quadatic natue assuming k γ Lemma Fo L and j integes thee holds (3) q L ( q; q) L ( j) j () q q q j L L j

5 POSITIVITY PRESERVING TRANSFORMATIONS 5 This coesponds to (4) f L, (q) ( q; q) L L which is about the simplest imaginable q-analogue of () Since the q-binomial coefficient on the ight is a polynomial with nonnegative coefficients 3 so is f L, (q) By the substitution q /q and the simple identities L a q q a(l a) L a q and we obtain the following coollay of Lemma Coollay Fo L and j integes thee holds (5) q (L ) L ( q; q)l ( j) j () This coesponds to () with k and γ 0 q, (a; q ) n ( ) n a n q (n ) (a ; q) n q q L L j Poof of Lemma Without loss of geneality we may assume that 0 j L Afte shifting + j the identity (3) coespond to (6) φ (q n, q n ; aq; q, aq n ) (a; q ) n (a; q) n with (a, n) (q j+, L j) (Thoughout this pape we denote the simultaneous vaiable changes a b,, a k b k by (a,, a k ) (b,, b k )) Equation (6) eadily follows fom the q-gauss sum 30, Eq (II8) (7) φ (a, b; c; q, c/ab) (c/a, c/b; q) (c, c/ab; q) Ou next esult is a somewhat moe complicated quadatic tansfomation, in accodance with () fo k and γ Lemma Fo L and j integes thee holds (8) ( + q L ) j () q 4 ( q + ; q ) L L ( j) q q 4 j L L j To make sense of the above lemma we need to extend ou ealie definition of the q-shifted factoial, and fo nonnegative n we set (a; q) n /(aq n ; q) n Note that this implies that /(q) n 0 With this definition it is once again clea that the coesponding polynomial f L, (q) has nonnegative coefficients Befoe poving (8) we state a vaiation that is not of the fom () Lemma 3 Fo L and j integes thee holds q 4 (+) L (9) + q ( q+ ; q ) L ( j) j () q q + q j 4 j(j+) L L j

6 6 ALEXANDER BERKOVICH AND S OLE WARNAAR Poof of Lemmas () and (3) Without loss of geneality we may assume that 0 j L Shifting + j the summations (8) and (9) coespond to a/b, q n, q n (0) 3φ aq, q n /b ; q, q (b; q) n(a; q ) n (a; q) n (b; q ) n with (a, b, n) (q j+, q j, L j) and (a, b, n) (q j+, q j+, L j), espectively Equation (0) follows fom the q q case of the q-pfaff Saalschütz sum 30, Eq (II4) witten in the fom () 3φ a, b, c d, abcq/d ; q, q povided the 3 φ teminates (q/d, abq/d, acq/d, bcq/d; q) (aq/d, bq/d, cq/d, abcq/d; q), Ou final solution to () povides a positivity peseving tansfomation of a quatic natue Lemma 4 Fo L and j integes thee holds L () q L ( q ; q ) L ( j) j () q q 4 Once again we state a vaiation that is not of the fom () Lemma 5 Fo L and j integes thee holds q L (3) + q ( q; q ) L ( j) j () q q4 qj L L j + q j L L j Poof of Lemma 4 Without loss of geneality we may assume that 0 j L Afte shifting + j the identity () coesponds to aq, aq 3, q n, q n (4) 4φ 3 a q, q 3 n, q 5 n ; q4, q 4 q n ( q; q) n( a; q ) n ( q ; q ) n ( a; q) n with (a, n) ( q j+, L j) The above equation follows fom 0, Eq () by the substitution (C, D, m, q) ( q n, aq, n/, q ) Unfotunately, the poof of 0, Eq () as stated in 0 appeas to be incomplete and below we povide the full details of the deivation of (4) Fist ecall Seas 4 φ 3 tansfomation 30, Eq (III5), which we wite in the fom a, b, c, d (5) 4φ 3 e, f, abcdq/ef ; q, q (q/f, abq/f, acdq/ef, bcdq/ef; q) a, b, e/c, e/d 4φ 3 (aq/f, bq/f, cdq/ef, abcdq/ef; q) e, abq/f, ef/cd ; q, q, povided both seies teminate Letting (a, b, c, d, e, f, q) (q n, q n, aq, aq 3, a q, q 3 n, q 4 ) (4) can be witten as aq, aq, q n, q n (6) 4φ 3 a q, q n, q 3 n ; q4, q 4 ( q; q) n( a; q ) n ( q; q ) n ( a; q) n

7 POSITIVITY PRESERVING TRANSFORMATIONS 7 At fist sight it may appea that little pogess has been made, but upon close inspection one may note that the paametes in this new 4 φ 3 seies ae tuned to allow the application of Singh s quadatic tansfomation 30, Eq (III) a, b, c, d (7) 4φ 3 abq /, abq /, cd ; q, q 4 φ 3 a, b, c, d a b q, cd, cdq ; q, q tue povided both seies teminate Indeed, utilizing this tansfom with (a, b, c, d, q) ((a/q) /, (aq) /, q n, q n, q ) we aive at (0) with (a, b) ( a, q) Equation (6) may also be deived fom the summation, Eq (43) with b (ediscoveed in 0, Eq ()) by making the substitutions (a, b, w, m, q) (aq,, aq (n+)/, n/, q ) Poof of Lemma 5 Without loss of geneality we may assume that 0 j L Afe shifting + j the sum (3) coespond to (6) with (a, n) ( q j+, L j) Ou final tansfomation fo q-binomial coefficients takes a fom that is slightly diffeent fom () Lemma 6 Fo L and j integes such that j L (mod ) thee holds L/3 q 3 4 (q 3 ; q 3 ) (8) (L ) ( q L ) (q 3 ; q 3 ) (q; q) (L 3) q 3 L 4 j ( j) q 3 (L 3j) j () When L 0 the facto multiplying the q-binomial coefficient in the summand on the left should be taken to be Poof of Lemma 6 Shifting + j and defining n (L 3j)/ we aive at the (a, b, c, d, q) (q n, q n, q n, q 3j+3, q 3 ) instance of () Again an impotant question is whethe the polynomial (9) f L, (q) (q3 ; q 3 ) (L ) ( q L ) (q 3 ; q 3 ) (q; q) (L 3) fo 0 3 L and L (mod ) has nonnegative coefficients To answe this is not entiely tivial and we need a genealization of a esult of Andews 0 that aose in connection with a monotonicity conjectue of Fiedman, Joichi and Stanton 6 Theoem Let k and n be positive integes, j {0,, n} and g gcd(n, j) Then A n,j,k (q) qk n q n j is a ecipocal polynomial of degee j(n j) + k n with nonnegative coefficients if k 0 (mod g) Fo k this is Andews esult 0, Thm Assuming the theoem it is not difficult to show that f L, (q) given by (9) is a polynomial with nonnegative coefficients Fist we note that fo 0 o 3 L this is obvious; f 3, (q) and f L,0 (q) ( + q L )(q 3 ; q 3 ) L /(q; q) L (L > 0), whee,

8 8 ALEXANDER BERKOVICH AND S OLE WARNAAR the positivity of the second polynomial follows fom ( q 3 )/( q) + q + q In the following we may theefoe assume 0 < 3 < L, which implies that k : gcd((l )/, ) (L 3)/ as follows Thee holds uk and (L )/ vk with v > u and gcd(u, v) Hence (L 3)/ (v u)k so that k (L 3)/ Next we obseve the decomposition f L, (q) ( qk )(q 3 ; q 3 ) (L 3)/ ( q 3k )(q; q) (L 3)/ ql q k A (L )/,,k(q 3 ), whee all thee factos on the ight ae polynomials with nonnegative coefficients The fist tem because k (L 3)/ so that ( q k )(q 3 ; q 3 (L 3)/ ) (L 3)/ ( q 3k ( + q j + q j ), )(q; q) (L 3)/ the second tem because k L, and the last tem because of Theoem with k g It is possible to aive at Theoem by modifying Andews poof fo k Instead, howeve, we will establish a moe geneal theoem genealizing esults of Haiman 33, 5 that he used to show polynomiality and nonnegativity of a conjectued expession fo a specialization of the Fobenius seies F(q, t) of diagonal hamonics Fo most of the teminology and notation used below we efe to 40, 49 Let s λ be the Schu function labelled by the patition λ and define j j k B λ,d,k (q) qk q d s λ(, q,, q d ) Theoem Let d and k be positive integes and λ a patition such that l(λ) d Set g gcd(d, λ ) Then B λ,d,k (q) is a ecipocal polynomial of degee k d + l(λ) i (d i)λ i with nonnegative coefficients fo evey λ if k 0 (mod g) Fo k this is due to Haiman Befoe poving the theoem let us show that it includes the pevious theoem as special case Fo notational convenience we set q δ (q d,, q, ) (δ (d,,, 0)) so that fo f a symmetic function f(, q,, q d ) may be witten as f(q δ ) Now we choose λ (j) and use that 40, Ch 3, Example, 49, Pop 9 j + d s (j) (q δ ) j Theefoe B (j),n j,k (q) qk q n j n qk j q n n A n,j,k (q) j By Theoem the statement of Theoem now follows, be it that j {0,, n } and g gcd(j, n j) Since gcd(n j, j) gcd(n, j) and since Theoem is tivially tue fo j n this completes ou deivation

9 POSITIVITY PRESERVING TRANSFORMATIONS 9 Poof of Theoem Let λ be the conjugate of the patition λ (λ,, λ d ) Then we have 40, Ch 3, Example, 49, Thm 7 (0) s λ (q δ ) q n(λ) x λ q d+c(x) q h(x) Hee fo each x (i, j) λ (a patition and its diagam ae identified) the hooklength and content of x ae given by h(x) λ i + λ j i j + and c(x) j i, espectively, and n(λ) d i (i )λ i To poceed futhe we need the following lemma, communicated to us by Richad Stanley Lemma 7 Let i d, and let ω i be an ith pimitive oot of unity l(λ) d, s λ (ωi δ ) 0 iff λ has a non-empty i-coe Then fo To pove this we note that i d and (0) imply that s λ (ωi δ ) 0 iff the numbe of hook-lengths h(x) divisible by i is stictly less than the numbe of contents c(x) divisible by i Next we ecall that the i-coe of λ is obtained fom λ by epeated emoval of bode stips of length i fom the diagam of λ until no futhe stips of length i can be emoved 40, Ch, Example 8(c), 49, Execise 759d It is staightfowad to veify that each time a bode stip is emoved, the numbe of hook-lengths and the numbe of contents divisible by i is deceased by one When we finally each the i-coe of λ the numbe of hook-lengths divisible by i becomes zeo On the othe hand, unless the i-coe is empty, thee will still be a content divisible by i, fo example, c(, ) 0 This completes the poof of the lemma Remak If i λ then λ has a non-empty i-coe If i λ and eithe λ o λ consists of a single ow, then λ has an empty i-coe Howeve, in geneal, the i-coe of λ is not necessaily empty when i λ Next, since s λ (q δ ) is a polynomial, the only potential poles of B λ,d,k (q) ae poles of R k,d (q) : ( q k )/( q d ) Clealy, R k,d (q) has fist ode poles at each ith pimitive oot of unity ω i, povided i >, i d, but i k Now, if k 0 (mod g), then i λ and, as a esult, the i-coe of λ is not empty Hence, by Lemma 7, s λ (ω δ i ) 0 Thus, if k 0 (mod g), evey pole of R k,d(q) is cancelled by a zeo of s λ (q δ ), and consequently B λ,d,k (q) is polynomial if k 0 (mod g) In the emainde we assume that k 0 (mod g) The degee of B λ,d,k (q) immediately follows fom the degee of s λ (q δ ) given in 40, Ch 3, Example To show that the polynomial B λ,d,k (q) has nonnegative coefficients and is ecipocal we use that s λ (q δ ) is a ecipocal, unimodal polynomial with nonnegative coefficients 40, Ch 3, Example, Ch 8, Example 4, 49, Execise 775c This immediately implies the ecipocality of B λ,d,k (q) To see that it also implies nonnegativity we denote the degee of s λ (q δ ) by D and note that it suffices to show positivity fo k g thanks to q k ( q g )( + q g + + q k g ) fo k mg Now, by the unimodality and nonnegativity of s λ (q δ ), it follows that ( q)s λ (q δ ) is a polynomial of degee D + with nonnegative coefficients up to the coefficient of q (D+)/ Hence B λ,d,g (q) + q + + qg q d ( q)s λ (q δ ) is a polynomial of degee D+g d with nonnegative coefficients up to the coefficient of q (D+)/ But by its ecipocality and by the fact that (D + )/ (D + g d)/ it follows that all its coefficients must be nonnegative

10 0 ALEXANDER BERKOVICH AND S OLE WARNAAR We conclude this section with the following emaks Remak Lemma 7 is closely elated to 40, Ch 3, Example 7(a) It is also a staightfowad coollay of 5, Lem Remak 3 It is impotant to ealize that B λ,d,k (q) can be a polynomial in q fo k 0 (mod g) Indeed, the agument given above suggests that B λ,d,k (q) is a polynomial as long as the i-coe of λ is not empty fo any i that divides d but not k Fo example, conside the 5-coe patition µ (5,,, ) Then B µ,d,k (q) q 9 q5 q q 7 q q k q q 4 q 9 q q 3 is a polynomial fo any positive k Note, howeve, that when λ (j), λ cannot have a non-empty i-coe if i j, i > Hence, B (j),d,k (q) A d+j,j,k (q) is a polynomial in q iff k 0 (mod g) Fo k this is due to Andews 0, Thm 3 Goup-like elations 3 Peliminaies Not all of the q-binomial tansfomations of the pevious two sections ae independent, and many elations of vaious degee of complexity can be found Such elations ae impotant because they often imply new summation o tansfomation fomulas Fo the esults of Section the ocuence of elations was fist investigated by Bessoud et al 0 and late studied in moe detail by Stanton 50 who intoduced the notion of the Bailey Roges Ramanujan goup Fo notational easons we wite q 8 γ f L, (q) in (9) as F L, (q) and add as a supescipt the elevant equation numbe Fo example, L, (q) q 4 (q; q) L (q; q) (L )/ (q; q) F () Likewise we wite q 4 γ f L, (q) in () as F L, (q) and again add equation numbes, and we wite F (8) L, (q) fo the kenel of (8) Fo instance, F (3) L, (q) q L ( q; q) L With this notation we quote fom 0, 50: q (3a) (3b) s () s () F () () (q)f F (7) () (q)f s, (q) F () (q), L, s, (q) F (6) (q), L, and the moe complicated (3a) s () F () () (q)f s, (q) s () F () () (q)f s, (q),

11 POSITIVITY PRESERVING TRANSFORMATIONS (3b) (3c) s () s () F (7) () (q)f s, (q) F (6) () (q)f s, (q) s () F (6) F () s () s () () (q)f s, (q) (q )F s, (7) (q), F () (q )F s, (6) (q), (3d) s () F (7) (q3 )F (8) s, (q) s () F (8) (q )F s, (7) (q) (It seems that (3a), (3b) and (3d) ae actually missing in 0, 50) The elations in equation (3) coespond to summations and the elations in (3) to tansfomations fo basic hypegeometic seies Fo example, afte shifting s s + and eplacing (L )/ by n, then using a polynomial agument to eplace q (+)/ by the indeteminate a, and finally using a polynomial agument to eplace q n by b, (3d) becomes the balanced tansfomation a, aq, b, b ω, b ω (33) 5φ 4 a, a, a q, b 6 q /a 4 ; q, q (a4 /b 6 ; q ) (a 3 ; q 3 ) (a 6 q 3 ; q 6 ) (a 4 ; q ) (a 3 /b 6 ; q 3 ) (a 6 q 3 /b 6 ; q 6 ) a, a q, a q, b 3, b 3 5 φ 4 a 3, a 3 q 3/, a 3 q 3/, b 6 q 3 /a 3 ; q3, q 3, povided both seies teminate, ie, povided a o b is of the fom q n Hee ω exp(πi/3) Fo a simple poof of (3) and (3), and hence fo a poof of the above new tansfomation we efe to the next (sub)section In the following we extend the analysis of 0, 50 and pesent two sets of elations, one of the type F F F as in (3) and one of the type F F F F as in (3) Especially the tansfomations implied by the second set ae inteesting as many appea to be new 3 Relations of the type F F F Ou fist set of esults, which should be ead as five diffeent ways to decompose F (3) (q), is given by (34) F (3) L, (q ) s () s () L, F () (q )F (6) s, (q 4 ) F (5) (q )F () s, (q 4 ) F () (q )F (5) s, (q ) s () F (6) () (q)f s, (q) F (8) (q )F () s, (q 4 )

12 ALEXANDER BERKOVICH AND S OLE WARNAAR Similaly, thee ae thee diffeent decompositions of F (8) L, (q), (35) F (8) L, (q ) s () s () F () (q )F (7) s, (q 4 ) F (5) (q )F () s, (q 4 ) F (7) () (q)f s, Poof Since all of the above eight esults (and those of Sections 3 and 33) aise in simila fashion we only show how to pove the vey fist elation Fist take (6) and make the substitution (L, q) (s, q ) Then multiply this by F () s0 s j () j () (q) and sum ove s to aive at s F () (q)f s, (6) (q ) ( j) q q j L s0 s j () F () (q) (q) s (s j) Now change the ode of summation on the left and apply () on the ight to get F () (q)f s, (6) (q ) q L j ( j) L j j () q s () Compaing this with (3) yields (36) ( j) j () q s () F () (q)f (6) s, (q ) F (3) L, (q) 0, which should hold fo all integes L and j such that 0 j L The above equation is of the fom h L, (q) 0, ( j) j () whee, without loss of geneality, it may be assumed that 0 j L Hence the lowe bound on the sum may be eplaced by j Recusively it can be seen that h L, (q) 0 is the unique solution Indeed, by taking j L and j L it follows that h L,L (q) h L,L (q) 0 Next taking j L and j L 3 it in tun follows that h L,L (q) h L,L 3 (q) 0 Repeatedly deceasing j by it thus follows afte L/ + steps that all h L, (q) fo 0 L must vanish Applying this easoning to (36) yields the desied F () (q)f s, (6) (q ) F (3) L, (q) 0 Like (3), the elations of (34) and (35) (which should all be ead as the lefthand side being equal to one of the ight-hand side expessions) imply summation fomulas The only one of these that is possibly new coesponds to the second elation of (35) Afte the eplacement (q +/, L ) (a, n) this sum can be stated as iq /, iq /, q n, q n (37) 4φ 3 q, a, q n ; q, q + a q n ( a q ; q 4 ) n /a + a q (a; q) n q 4

13 POSITIVITY PRESERVING TRANSFORMATIONS 3 Pesumably this follows by contiguity fom the b iq / case of the easily established b, q/b, q n, q n 4φ 3 q, a, q n /a ; q, q (ab, aq/b; q ) n (a; q) n o fom (37) on page 6 with bq In Section 6 we edeive (37) fom the moe geneal tansfomation fomula (67) 33 Relations of the type F F F F This time thee ae athe a lage numbe of esults, all of which can be poved using the method detailed in Section 3 Those elations that imply base-changing tansfomations fom q to q k fo fixed k have been gouped togethe Hee k will be an element of the set {, 4/3, 3/,, 4, 6, 9, } 33 Linea tansfomations Thee ae just two linea elations (38a) (38b) L/ L/ L/ F (5) (3) (q)f s, (q ) L/ F (3) (5) (q)f s, (q ) F (8) (5) (q)f s, (q ), F (8) (3) (q)f s, (q ), which ae dual in the sense of q /q The coesponding q-hypegeometic tansfomations ae nothing but specializations of the identity obtained by equating the ight-hand sides of the Jackson tansfomations 30, Eq (III4) and 30, Eq (III5) 33 Tansfomations fom q to q 4/3 Much moe inteesting than (38) ae the genealized commutation elations L/3 L/3 F (8) (3) (q)f s, (q 3 ) F (8) () (q)f s, (q 3 ) s3 s () s3 s () F (3) (q)f (8) s, (q 4 ), F () (q)f (8) s, (q 4 ) Making the vaiable change s s + on the left and s s + 3 on the ight and then substituting (q 6, L 3) (a, n), the above elations imply the balanced and almost balanced fomulas iq 3/, iq 3/, q n, q n, q n 5φ 4 q 3, a / q 3/, a / q 3/, q 3 3n /a ; q3, q 3 ( q, a; q ) n ( q; q) n (a; q 3 ) n 5φ 4 and iq 3/, iq 3/, q n, q n, q n 5φ 4 q 3, a / q 3/, a / q 3/, q 3 3n /a ; q3, q 6 q n ( q, a; q ) n ( q; q) n (a; q 3 ) n 6φ 5 a /3, a /3 ω, a /3 ω, q n, q n a, aq, q n, q 3 n ; q 4, q 4 a /3, a /3 ω, a /3 ω, aq 4, q n, q n a, a, aq, q 3 n, q 5 n ; q 4, q 4,

14 4 ALEXANDER BERKOVICH AND S OLE WARNAAR espectively To the best of ou knowledge these ae the fist examples of a tansfomations elating base q 3 and q Tansfomations fom q to q 3/ Again thee ae two esults, not dissimila to the pevious pai; L/3 L/3 F (8) (8) (q)f s, (q 3 ) F (8) (9) (q)f s, (q 3 ) s3 s () s3 s () F (8) (8) (q)f s, (q ), F (9) (8) (q)f s, (q ) Making the same vaiable change as above and then substituting ( q 3, L 3) (a, n), this yields and a /3, a /3 ω, a /3 ω, q n, q n 5φ 4 a, a, aq, q n ; q, q /a (a ; q 3 ) n (a; q ) n ( a; q) n 5φ 4 a /3, a /3 ω, a /3 ω, q n, q n 5φ 4 aq, aq, a, q n ; q, q /a a q n a (a ; q 3 ) n (aq; q ) n ( aq; q) n Both these esults should be compaed with (33) a /, a /, q n, q n, q n a, aq 3/, aq 3/, q 3 3n /a ; q3, q 3 5 φ 4 a / q 3/, a / q 3/, q n, q n, q n aq 3, aq 3/, aq 3/, q 3 3n /a ; q 3, q Quadatic tansfomations Thee ae quite a numbe of diffeent elations of a quadatic natue Fist, (39a) (39b) F () (5) (q)f s, (q) F () (8) (q)f s, (q) F () (3) (q)f s, s () s () s () (q) F (3) () (q)f s, (q ), F (8) () (q)f s, (q ) F (3) () (q)f s, (q )

15 POSITIVITY PRESERVING TRANSFORMATIONS 5 The fist equality in (39a) coesponds to a specialization of the tansfomation 30, Eq (III4), and the second equality implies the (a, b, c, n) (q +/,, 0, L ) specialization of b, c, c, q n (30) 4φ 3 a, c, bq n /a ; q, q (a /b; q) n (c ; q ) n (a, c, a/b; q) n 4φ 3 a /b, a /c, q n, q n a /b, a q/b, q n /c ; q, q, poved in Section 6 Similaly, the second equality in (39b) coesponds to a specialization of the tansfomation 30, Eq (III), and the fist equality implies the (a, b, c, n) (q +/,, iq /, L ) specialization of (30) Next ae the fou closely elated esults (3a) (3b) (3c) L/ L/ L/ L/ F (3) (3) (q)f s, (q ) L/ F (5) (3) (q)f s, (q ) L/ F (3) () (q)f s, (q ) F (3) (3) (q)f s, (q 4 ), F (3) (5) (q)f s, (q 4 ), F () (3) (q)f s, (q 4 ), (3d) L/ L/ F (5) () (q)f s, (q ) F () (5) (q)f s, (q 4 ), The fist as well as the last two elations ae dual in the sense of q /q Afte the substitution (q 4+, L ) (a, n) equation (3a) implies iq, iq, q n, q n 4φ 3 q, a /, a / ; q, aq n ( q; q ) n 0, aq, q n, q n 4φ 3 ( q; q) n a, q n, q 3 n ; q4, q 4 and equation (3c) implies iq, iq, q n, q n 4φ 3 q, a /, a / ; q, aq n+ Finally thee holds q n ( q ; q ) n 0, aq, q n, q n 4φ 3 ( q; q) n a, q 3 n, q 5 n ; q4, q 4 (3a) (3b) L/ L/ L/ F (8) (3) (q)f s, (q ) L/ F (8) () (q)f s, (q ) F (3) (8) (q)f s, (q 4 ), F () (8) (q)f s, (q 4 )

16 6 ALEXANDER BERKOVICH AND S OLE WARNAAR Afte the eplacement (q, L ) (a, n) these yield iq, iq, q n, q n (33) 4φ 3 q, aq, q n /a ; q, q and iq, iq, q n, q n (34) 4φ 3 q, aq, q n /a ; q, q 4 ( a; q) n( q; q ) n ia, ia, q n, q n ( q; q) n ( a; q 4φ 3 ) n a q, q n, q 3 n ; q4, q 4 q n ( q ; q ) n ( a; q) n ( q; q) n ( a; q ) n 5φ 4 ia, ia, a q 4, q n, q n a, a q, q 3 n, q 5 n ; q4, q 4, espectively It is not had to see that (33) is a special case of b, q /b, q n, q n (35) 4φ 3 q, aq, q n /a ; q, q ( a; q) n( q; q ) n aq/b, ab/q, q n, q n ( q; q) n ( a; q 4φ 3 ) n a q, q n, q 3 n ; q4, q 4, which genealizes (6) and follows by fist applying Singh s quadatic tansfomation (7) to the ight-side and then using Seas 4 φ 3 tansfomation (5) (Equation (35) also follows fom, Eq (43) by a single use of Seas tansfom) Because of the 5 φ 4 seies on the ight, it is unclea whethe (34) admits a simila kind of genealization 335 Quatic tansfomations Ou list of quatic elations begins with (36a) (36b) F (6) (5) (q)f s, (q) F (7) (8) (q)f s, (q) F (7) (3) (q)f s, s () s () (q) F (8) (q )F (6) s, (q ), F (8) (q )F (7) s, (q ) The fist equality in (36a) once again coesponds to a specialization of 30, Eq (III) Moe inteesting ae the second equality in (36a) and the genealized commutation elation (36b) These pove the (a, b, n) (q +/, 0, L ), espectively, (a, b, n) (q +/, q, L ) case of b /, b /, q n, q n (37) 4φ 3 a, b, q n ; q, q /a + a q n + a q ( a q ; q 4 ) n (a; q) n 4φ 3 bq, bq 3, q n, q n a q 3, b q, q 5 4n /a ; q4, q 4,

17 POSITIVITY PRESERVING TRANSFORMATIONS 7 established in Section 6 As a vaiation on the above thee also holds (38a) (38b) F (6) (3) (q)f s, (q) F (6) (8) (q)f s, (q) s () s () F (3) (q )F (6) s, (q ), F (3) (q )F (7) s, (q ) Hee (38a), espectively, (38b) imply the (a, b, n) (q +/, 0, L ) and (a, b, n) (q +/, q, L ) instances of b /, b /, q n, q n (39) 4φ 3 ; q, a q n a, a, b ( a q; q ) n (a ; q ) n 4φ 3 bq, bq 3, q n, q n a q, a q 3, b q ; q 4, a 4 q 4n Once again this is poved in Section 6 By making the substitution (q +/, L ) (a, n) the elation (30) F (7) (5) (q)f s, (q) s () F (5) (q )F (6) s, (q ), yields the b limit of the quatic tansfomation b /, b /, q n, q n (3) 4φ 3 a, b, q n ; q, q /a ( a q; q ) n (b ; q 4 ) n (a; q) n (b ; q ) n 4φ 3 a /b, a q /b, q n, q n a q, a q 3, q 4 4n /b ; q4, q 4 It is not had to pove this identity by applying Seas 4 φ 3 tansfomation (5) with (a, b, c, d, e, f, q) (q n, q n, bq, bq 3, a q 3, b q, q 4 ) to the ight-hand side of (37) Next is the pai F () (3) (q)f s, (q) F () () (q)f s, (q) s () s () F (3) (q)f () s, (q 4 ), F () (q)f () s, (q 4 ) Afte the substitutions (q +/, L ) (a, n) these lead to (3) iq /, iq /, q n 3φ ; q, aq n ( q; q ) n a q, q n, q n 3φ q, a ( q, a; q) n q n, q 3 n ; q 4, q 4

18 8 ALEXANDER BERKOVICH AND S OLE WARNAAR and iq /, iq /, q n (33) 3φ ; q, aq n+ q, a q n ( q ; q ) n a q 3, q n, q n 3φ ( q, a; q) n q 3 n, q 5 n ; q4, q 4, which we failed to genealize to the level of 4 φ 3 (o 5 φ 4 ) seies It is howeve not had to see that by applying Singh s tansfomation (7) to the ight-hand side, (3) becomes the (b, c) (, iq / ) limit of (30) It is also possible to aive at (3) and (33) (with A eplaced by a) by taking the a, b limit in (35) and (36) such that A bq n /a is fixed, and by then tansfoming the esulting 3 φ seies on the ight using 30, Eq (III3) Ou last two quatic commutation elations ae athe inteesting, (34a) (34b) F () (3) (q)f s, (q) F () () (q)f s, (q) s () s () F (3) (q)f () s, (q 4 ), F () (q)f () s, (q 4 ) Equation (34a) implies the (a, b, n) (q +,, L ) instance of iq /, iq /, b /, b /, q n (35) 5φ 4 q, a /, a /, bq n /a ; q, q ( q, a /b; q ) n ( q, a/b; q) n (a; q ) n 5φ 4 a /b, aq, aq, q n, q n a /b, a q /b, q n, q 3 n ; q4, q 4 This esult, which will be poved in Section 6, simplifies to (6) fo b Similaly, (34b) coesponds to the (a, b, n) (q +,, L ) case of iq /, iq /, b /, b /, q n (36) 5φ 4 q, a /, a /, bq n ; q, q /a q n ( q, a /b; q ) n ( q, a/b; q) n (a; q ) n 5φ 4 a /b, aq, aq 3, q n, q n a /b, a q /b, q 3 n, q 5 n ; q4, q 4 When b this simplifies to (4) and when aq (and b b ) to (37) 3φ (b, b, q n ; q, b q n ; q, q ) q n (q /b ; q ) n ( q, q /b ; q) n needed shotly The poof of (36) can again be found in Section 6 Both (35) and (36) may be futhe manipulated into new quadatic tansfomations as follows The left-hand side of (35) simplifies to a 3 φ seies by the (b, x, y) (i(bq/a) /, (a/b) /, i(a/q) / ) case of 30, Eq (35); a q n (38) bx, bx, by, by, q n 5φ 4 q, bxy, bxy, b ; q, q q n (q /b ; q ) n x, y, q n ( q, q/b 3φ ; q) n b x y, b q n ; q, b q 3

19 POSITIVITY PRESERVING TRANSFORMATIONS 9 Afte the futhe substitution (a, b, q) (aq, aq/b, q) this leads to (39) a, aq, b, q n, q n 5φ 4 abq, abq 3, q n, q 3 n ; q4, q 4 (aq, bq; q ) n a, b, q n (q, abq; q 3φ ) n aq, q n /b ; q, q b When (b, q) ( aq, q) the left is summable by (6) and we infe the futhe identity (330) φ (a, q n ; q n /a; q, q/a) ( q, aq; q) n (aq ; q ) n, which also follows fom 30, Execise 8 equation (39) may also be stated as a, aq, b, c, cq 5φ 4 abq, abq 3, cq, cq 3 ; q4, q 4 By the usual polynomial agument (q, cq/a, cq/b, q/ab; q ) (cq, q/a, q/b, cq/ab; q ) 3φ a, b, c aq, cq/b ; q, q, b povided both seies teminate Fo c aq the 3 φ seies on the ight becomes a φ which pecisely takes the fom of the sum side of the Bailey-Daum summation 30, Eq (II9) Remak 3 When a q j with j, equation (330) may be put in the fom n k + j n k + j n + j (33) q k k j n k0 q q This has the following elegant patition theoetic intepetation The expession k + j q k k is the geneating function of patitions of exactly k pats, with all pats being odd and no pats exceeding j The expession n k + j j is the geneating function of patitions of at most n k pats, with all pats being even and no pats exceeding j Hence the summand on the left of (33) is the geneating function of patitions of at most n pats, with no pats exceeding j and exactly k odd pats When summed ove the numbe of odd pats this gives the geneating function of patitions of at most n pats with no pats exceeding j, in accodance with the ight-hand side of (33) To also ewite (36) as a quadatic tansfomation equies a bit moe wok Indeed, in ode to tade the 5 φ 4 on the left fo a 3 φ we need to pove the following companion to (38): bx, bx, by, by, q n (33) 5φ 4 q, bxy, bxy, b ; q, q q n q q q n (q /b ; q ) n ( q, q /b ; q) n 3φ x, y, q n b x y, b q 4 n ; q, b q 3

20 0 ALEXANDER BERKOVICH AND S OLE WARNAAR Using this with (b, x, y) (i(b/aq) /, (a/b) /, i(aq) / ) and making the futhe substitution (a, b, q) (aq, aq /b, q) yields a, aq, b, q n, q n (333) 5φ 4 abq, abq, q 3 n, q 5 n ; q4, q 4 (aq, bq ; q ) n (q, abq ; q 3φ ) n a, b, q n aq, q 3 n /b ; q, q b When (a, b, q) (aq, a, q) the sum on the left can be caied out by (4) leading to (334) φ (aq, q n ; q 3 n /a; q, q /a) q n ( q, a; q) n (a/q; q ) n This sum, which is in fact (330) with ode of summation evesed, will be needed in Section 8 Again we may eplace q n in (333) by c to find a, aq, b, c, cq 5φ 4 abq, abq, cq 3, cq 5 ; q4, q 4 (q3, cq 3 /a, cq 3 /b, q 3 /ab; q ) (cq 3, q 3 /a, q 3 /b, cq 3 /ab; q ) 3φ a, b, c aq, cq 3 /b ; q, q, b povided both seies teminate To the best of ou knowledge (39) and (333) ae new, and the esult closest to these tansfomations that we wee able to obtain using just elementay esults fom 30 is 4φ 3 a, aq, q n, q n b q, aq n /b, aq n /b ; q, q (b ; q ) n b, q n (b φ, b/a; q) n q n /b ; q, q a This genealizes 30, Execise 6 (i) obtained when a tends to 0, and follows fom 30, Execise 34 and 30, Eq, (III8) Poof of (33) Take (37) and let j be the summation vaiable in the 3 φ seies Replace n by n k, shift j j k and multiply both sides by (x, y ; q ) k (q, b x y ; q ) k (q n ; q) k (b q n ; q) k (bq) k Next sum k fom 0 to n and intechange the ode of the sums ove j and k on the left This gives, afte some tedious but elementay manipulations involving q-shifted factoials, n x, y, q j (b 3φ b x y, q j /b ; q, q ; q ) j (q n ; q) j (q ; q ) j (b q n q j ; q) j j0 q n (q /b ; q ) n x, y, q n ( q, q /b 3φ ; q) n b x y, b q 4 n ; q, b q 3 The 3 φ can be summed by () esulting in (33)

21 POSITIVITY PRESERVING TRANSFORMATIONS 336 Sextic tansfomations Both ou esults take the fom of genealized commutation elations Fist, F (8) (8) (q)f s, (q) F (8) (q3 )F s, (8) (q ), s () which, by the substitution ( q, L ) (a, n), yields a /, a /, q n, ωq n, ω q n 5φ 4 a, aq /, aq /, q 3n /a ; q, q Second, L/3 s () a3 q 3n a 3 (a 3 ; q 6 ) n ( a 3 q 3 ; q 3 ) n (a q; q) 3n F (8) (q )F (7) s, (q 3 ) 5 φ 4 a q, a q 4, a q 6, q 3n, q 3 3n a 3 q 3, a 3 q 6, a 3 q 6, q 6 6n /a 3 ; q6, q 6 s3 s () F (7) (8) (q)f s, which, by the substitution (q 3, (L 3)/) (a, n), yields a /3, a /3 ω, a /3 ω, q n, q n 5φ 4 a, a, aq /, q / n ; q, q /a a4 q 6n a 4 (aq / ; q) n (a q ; q ) n (a 4 ; q 6 ) n (q), 5 φ 4 aq 3/, aq 9/, q n, q n, q 4 n a q 3, a q 3, a q 6, q 6 6n /a 4 ; q6, q Tansfomation fom q to q 9 As ou second-last last elation thee holds s3 s () F (8) (8) (q)f s, (q) L/3 s () Afte eplacing (q 3, (L 3)/) (a, n) this becomes a /3, a /3 ω, a /3 ω, q n, ωq n, ω q n 6φ 5 a, a, aq /, aq /, q 3n /a ; q, q a6 q 6n a 6 (a 6 ; q 6 ) n (a q; q) 3n 6φ 5 F (8) (q 3 )F (8) s, (q 3 ) a q 3, a q 6, a q 9, q 3n, q 3 3n, q 6 3n a 3 q 9/, a 3 q 9/, a 3 q 9, a 3 q 9, q 9 9n /a 6 ; q9, q 9 To the best of ou knowledge this is the fist tansfomation between the bases q and q Tansfomation fom q to q Also ou vey last elation is an isolated esult because F (3) commutes with all but F (8) ; F (8) () (q)f s, (q) F () (q 3 )F s, (8) (q 4 ) s ()

22 ALEXANDER BERKOVICH AND S OLE WARNAAR Making the eplacement (q +, L ) (a, n) this coesponds to iq /, iq /, q n, ωq n, ω q n 5φ 4 q, a /, a /, q 3n ; q, q /a q 3n ( q 3, a 3 q 3 ; q 6 ) n ( q 3 ; q 3 ) n (a; q) 3n 5φ 4 a q, a q 6, a q 0, q 6n, q 6 6n a 3 q 3, a 3 q 9, q 9 6n, q 5 6n ; q, q We believe this to be the fist example of a tansfomation elating base q to base q 4 Invese tansfomations 4 Main esults When iteating any of the tansfomations of Section it is often impotant to stat with an as simple as possible q-binomial identity as seed One possible way to detemine whethe a potential seed can actually be educed is by applying the inveses of the tansfomations of Lemmas 6 Fo a tansfomation of the type () we conside a fomula of the fom L q 4 L γj q 4 γl f L, (q) j (L j) q k χ(l j ()) as its invese Hee χ is the tuth function; χ(tue) and χ(false) 0 Indeed, eplacing (L, ) (, s) in () and then using this to eliminate the q-binomial coefficient in the above summand yields s L s0 s j () q 4 γ(s L ) (s j) q k s f L, (q)f,s (q) This is obviously satisfied if the invese elations (4a) f L, (q)f,s (q) δ, (4b) s f L, (q) f,s (q) δ s (L j) q k χ(l j ()) hold Hee the second equation follows fom the fist and the fact that f L, (q) is nonzeo if and only if 0 L Invese elations like (4) have been much studied in the theoy of basic hypegeometic seies Most impotantly, they ae elated to the Bailey tansfom 5, 9, 3, 7, 8, 54, the poblem of q-lagange invesion 3, 3 and summations and tansfomations of q-hypegeometic seies,, 3, 4, 7, 38, 44 The fist invese is that of Lemma Lemma 4 Fo L and j integes thee holds q L L ( ) +L q (L ) L ( q; q)l q j q j L χ(l j ()) (L j) q

23 POSITIVITY PRESERVING TRANSFORMATIONS 3 Poof All we need to do is show that f L, (q) ( ) +L q (L ) ( q; q)l L and f L, (q) as given by (4) satisfy (4a) Shifting + s this becomes the n L s case of φ 0 (q n ; ; q, q) δ n,0, which follows fom the q-binomial theoem 30, Eq (II4) (4) φ 0 (q n ; ; q, z) (zq n ; q) n Altenatively we can pove Lemma 4 without esoting to invese elations Assuming 0 j L and shifting + j the identity of the lemma becomes the (a, c, n) (0, q j+/, L j) instance of 30, Eq (II7) a q, c, c, q n (43) 4φ 3 c, aq n/ ; q, q, aq n/ due to Andews 4 Next is the invese of Lemma Lemma 4 Fo L and j integes thee holds q 4 L L q (q, c /a ; q ) n/ (c q, /a ; q ) n/ χ(n 0 ()) ( ) +L q (L ) L ( q L+ ; q ) L j q L 4 j χ(l j ()) (L j) q Poof Using that (4) emains unchanged if we multiply f L, (q) by x (q)y L (q) and divide f L, (q) by x L (q)y (q) (x (q) 0, y L (q) 0) we this time need to show that f L, (q) ( + q L ) ( q+ ; q ) L (q; q) L, f L, (q) ( ) +L q (L ) ( q L+ ; q ) L (q; q) L satisfies (4a) Shifting +s this is (43) with c a and (a, n) ( q s, L s) Altenatively, we may assume 0 j L and shift + j to find that Lemma 4 is (43) with (a, c, n) ( q j, q j+/, L j) The following lemma, coesponding to the invese of (9) is (liteally) the odd one out as the sum on the left does not vanish when L j is odd Lemma 43 Fo L and j integes such that j L (mod ) thee holds q 4 L(L+) ( + q L ) ( ) +L q (L ) L ( q L+3 ; q ) L j q 4 j(j+) ( + q j L ) (L j) q

24 4 ALEXANDER BERKOVICH AND S OLE WARNAAR Poof The diffeence with the pevious two cases is that the pai f L, (q) ( q+ ; q ) L (q; q) L, f L, (q) ( ) +L q (L ) ( q L+3 ; q ) L (q; q) L only satisfies (4) fo s L (mod ) Indeed, shifting + s and substituting the above, (4a) becomes a, bq, c, c, q n (44) 5φ 4 c, b, aq n/ ; q, q, aq n/ a bq n (q, c /a ; q ) n/ a q n b (c q, /a ; q if n is even, ) n/ q n a b (q, c q/a ; q ) (n )/ a q n b (c q, q/a ; q if n is odd, ) (n )/ with c a, b 0 and (a, n) ( q s+, L s( 0 ())) Note in paticula that fo this choice of a, b and c the ight side of (44) only tivializes to δ n,0 fo even values of n, explaining why L s must be even The poof of (44) is given in the next subsection Also the diect poof of the lemma elies on a special case of (44) Assuming 0 j L and shifting + j Lemma 43 is (44) with (a, b, c, n) ( q j+, 0, q j+/, L j ( Z)) The inveses of the two quatics tansfoms () and (3) ae as follows Lemma 44 Fo L and j integes thee holds L ( ) +L ( q; q ) L j Lemma 45 Fo L and j integes thee holds ( + q L ) q L ( ) +L q ( q ; q ) L q L χ(l j ()) (L j) q 4 j q j ( + q j ) L χ(l j ()) (L j) q 4 Poof The Lemmas 44 and 45 follow fom () and (3) and the a q and a q instances of the invese pai (45a) f L, (q) a (a; q ) L (q ; q ) L, (45b) f L, (q) (/a) (/a; q ) L (q ; q ) L Shifting + s in equation (4) this follows fom the n L s case of (46) φ (a, q n ; aq n ; q, q ) δ n,0, which is a specialization of (7)

25 POSITIVITY PRESERVING TRANSFORMATIONS 5 The diect poof of Lemmas 44 and 45 is only inteesting fo the latte Namely, if we assume that 0 j L and shift + j then Lemma 44 is equation (43) with (a, c, n) ( q j L, q j+/, L j), but Lemma 45 is c, c, bq, q n, q n (47) 5φ 4 c q, b, iq 3/ n ; q, q, iq3/ n (q ; q 4 ) n/ ( c q ; q ) n (c 4 q ; q 4 ) n/ ( q ; q if n is even, ) n q c q c b b with b c and (c, n) (q j+/, L j) Section 4 (q 6 ; q 4 ) (n )/ ( c q; q ) n (c 4 q 6 ; q 4 ) (n )/ ( q; q ) n if n is odd, The identity (47) will be poven in Remak 4 By (7) it can also be shown that (4) with (45) (nomalized) is the b /a case of M(a)M(b) M(ab), with M(a) the infinite-dimensional, lowe-tiangula matix M(a) (M i,j (a)) i,j 0 whose enties ae given by i M i,j (a) a j (a; q ) i j j q Finally we state the invese of the cubic tansfomation of Lemma 6 Lemma 46 Fo L and j integes such that j L (mod ) thee holds q 3 4 L 3 L j () ( ) (+L) q ( (3L ) ) (q 3 ( L+) ; q 3 ) (q; q) (3L ) (3L ) ( 3j) q 3 4 j L (L j) Poof This case is quite diffeent fom the pevious ones in that f L, (q) coesponding to (8) is nonzeo if and only if 0 3 L As a consequence only a left-invese exists, and we claim that f L, (q) (aq3 ; q 3 ) (L ) ( aq L ) (aq 3 ; q 3 ) (q; q) (L 3) f L, (q) ( ) (+L) q ( (3L ) ) (aq 3 ( L+) ; q 3 ) (3L ) (q; q) (3L ) with L (mod ) satisfies (48) 3 3s 0 () f L, (q)f,s (q) δ fo s L (mod ) Note that this suffices to conclude Lemma 46 fom (8) by taking a To pove that (48) indeed holds we epace + 3s to aive at the (b, n) (aq 3s, 3(L s)/( 0 (3))) case of (49) n bq b (b; q 3 ) (q n ; q) q (q; q) (bq 3 n ; q 3 ) δ n,0, q 3

26 6 ALEXANDER BERKOVICH AND S OLE WARNAAR which is 30, Eq (367); p q 3, a 0 due to Bessoud 8, Gaspe 7 and Kattenthale 38 Fo a diect poof of the lemma we shift + j to obtain the singula case (c, n) (q 3j+, 3(L j)/( 0 (3))) of (40) n (c; q) (q n ; q) q (q, c; q) (cq n ; q 3 (q, q ; q 3 ) 3 n ) (q/c, cq ; q 3 χ(n 0 (3)), ) 3 n which is 3, Eq (43); k n k, A q n /c of Gessel and Stanton 4 Poofs of (44) and (47) Befoe poving the what-we-believe-to-be new balanced 5 φ 4 sum (44) we note that Andews identity (43) aises as the case b a (o b q n ) Since (43) povides a q-analogue of Watson s 3 F summation, (44) also povides a genealization of Watson s sum Specifically, eplacing (a, b, c) (q a/, q b, q c ) in (44) and then letting q tend to one we find (4) a(b + n) ( a, b +, c, n, c a) n/ b(a + n) (c + 4F 3 c, b,, a) if n is even, n/ (a n + ); n(b a) (, c a + ) (n )/ b(a + n) (c +, a) if n is odd, (n )/ whee we employ standad notation fo hypegeometic seies, 30, 48 Fo b a this yields Watson s (teminating) 3 F sum (Whipple extended Watson s esult to nonteminating seies, but at the 4 F 3 level this no longe appeas to be possible) At the end of this section anothe extension of Watson s sum is be given Poof of (44) It is not had to establish (44) by application of the contiguous elation 37, Eq (38) aq, b, c, (A) ( b)(a c) a, bq, c, (A) (4) φ s ; q, z (B) ( a)(b c) φ s ; q, z (B) ( c)(a b) a, b, cq, (Aq) ( a)(b c) φ s ; q, z (B) Hee (A), (B) and (Aq) ae shothand notations fo a,, a 3 and b,, b s and a q,, a 3 q, espectively Utilizing (4) with (a, b, c) (b, a, q n ), the lefthand side of (44) tansfoms into the sum of two b-independent 4 φ 3 seies Both ae summable by (43) to yield the desied ight-hand side Poof of (47) To show (47) we split its left-hand side by (4) with (a, b, c) (b, q n, q n ) so that LHS(47) ( + qn )( bq n ) q n ( b) c, c, q n, q n 4φ 3 c q, iq 3/ n ; q, q, iq3/ n ( qn )( + bq n ) c, c, q n, q n q n 4φ 3 ( b) c q, iq 3/ n ; q, q, iq3/ n

27 POSITIVITY PRESERVING TRANSFORMATIONS 7 Both the 4 φ 3 seies on the ight ae summable by a, c, c, q n (43) 4φ 3 c q, aq n/ ; q, q, aq n/ c a q n (q; q ) n/ (c q /a ; q ) n/ a q n (c q; q ) n/ (q /a ; q if n is even, ) n/ a c q n (q; q ) (n+)/ (c q/a ; q ) (n )/ a q n (c q; q ) (n+)/ (q/a ; q if n is odd, ) (n )/ leading to the ight side of (47) To complete the poof we need to deal with (43) By 37, Eq (3) (A) (A) +φ aq, (B) ; q, z + φ a, (B) ; q, z + i ( A i) az ( a)( aq) i ( B i) + φ (Aq) aq, (B) ; q, z with a c q the left side of (43) can be witten as the sum of two 4 φ 3 seies, both of which can be summed by the b limit of (44) This esults in the ight side of (43) To conclude this section we wish to point out that (44) is cetainly not the only genealization of (43) that may be obtained using contiguous elations Fo example, by (43) and 37, Eq (33) a, (A) +φ b, (B) ; q, z + φ a/q, (A) b/q, (B) ; q, z z(a b) i + ( A i) (q b)( b) i ( B i) + φ a, (Aq) bq, (Bq) ; q, z with (a, b, (A), (B)) (bq, c q, (a q, c, c, q n ), (b, aq n/, aq n/ )) it follows that a q, bq, c, c, q n (44) 5φ 4 c q, b, aq n/ ; q, q, aq n/ (q, c /a ; q ) n/ (c q, /a ; q if n is even, ) n/ c b a q (q, c q /a ; q ) (n+)/ b c a q (c q, q /a ; q if n is odd ) (n+)/ Fo b c this simplifies to (43) and fo (a, b, c) (q a/ /, q b, q c ) togethe with q it yields ( a, b +, c, n, c a + ) n/ (c + 4F 3 c +, b,, a) if n is even, n/ (a n + ); a(b c) (, c a) (n+)/ b(a c) (c +, a) if n is odd (n+)/ This is to be compaed with (4) Fo b c this is again Watson s 3 F sum

28 8 ALEXANDER BERKOVICH AND S OLE WARNAAR Finally we emak that othe balanced 4 φ 3 summations than (43) follow fom (44) and (44) Taking b c /q in (44) and b a q in (44) leads to two moe such esults Especially the latte is appealing as some factos on the left of (44) nicely cancel leading to (q, c q /a ; q ) n/ a, c, c, q n (c q, q /a ; q if n is even, ) n/ 4φ 3 c q, aq n/ ; q, q, aq n/ (q, c q/a ; q ) (n+)/ (c q, q/a ; q if n is odd, ) (n+)/ whee we have also eplaced a by a/q 5 The Bailey lemma As alluded to in the intoduction, the q-binomial tansfomations of the fist two sections ae closely elated to Bailey s lemma Pesently we will make this moe pecise and estate ou esults in tems of tansfomations on Bailey pais Fist we ecall the definition of a Bailey pai 6 If α(a; q) {α L (a; q)} L 0 and β(a; q) {β L (a; q)} L 0 ae sequences such that β L (a; q) α (a; q) (q; q) L (aq; q) L+, then (α(a; q), β(a; q)) is called a Bailey pai elative to a and q The Bailey lemma is the following poweful mechanism fo geneating new Bailey pais 6, 7, 9, 3, 4, 54 Lemma 5 If (α(a; q), β(a; q)) is a Bailey pai elative to a and q, then so is (α (a; q), β (a; q)) given by (5a) (5b) α L(a; q) (b, c; q) L(aq/bc) L α L (a; q), (aq/b, aq/c; q) L β L(a; (aq/bc; q) L (b, c, q L ; q) q q) (q, aq/b, aq/c; q) L (bcq L β (a; q) /a; q) Fo (b, c) (, ) and (b, c) (, (aq) / ) this is equivalent to () and (), espectively Befoe we state simila such esults aising fom the tansfomations of section we ecall the base-changing Bailey-pai tansfomations of Bessoud et al 0 The fist esult is (equivalent to) 0, Thm Lemma 5 If (α(a; q), β(a; q)) is a Bailey pai elative to a and q, then the pai (α (a ; q ), β (a ; q )) given by α L(a ; q ) (b; q) ( L aq ) L( ) (5a) L q ) (L αl (a; q), (aq/b; q) L b β L(a ; q ( aq/b; q) L (b; q) (q L ; q ) q (5b) ) ( aq; q) L (q, a q /b ; q ) L ( bq L β (a; q) /a; q) foms a Bailey pai elative to a and q Fo b 0, b and b (aq) / this yields the equations (E), (E) and Eq (E3) of 50 By some simple vaiable changes, (E) and (E3) can be seen to be equivalent to (6) and (7)

29 POSITIVITY PRESERVING TRANSFORMATIONS 9 The next esult is (equivalent to) 0, Thm 3, 50, Eq (T) and (8) Lemma 53 If (α(a; q), β(a; q)) is a Bailey pai elative to a and q, then the pai (α (a 3 ; q 3 ), β (a 3 ; q 3 )) given by (53a) (53b) α L(a 3 ; q 3 ) a L q L α L (a; q), β L(a 3 ; q 3 (aq; q) 3L ) (q 3 ; q 3 ) L (a 3 q 3 ; q 3 ) L foms a Bailey pai elative to a 3 and q 3 (q 3L ; q 3 ) q (q 3L /a; q) β (a; q) To the above thee lemmas we now add seveal new base-changing Bailey lemmas Fist is a Bailey-type lemma of a quadatic natue Lemma 54 If (α(a; q), β(a; q)) is a Bailey pai elative to a and q, then so is (α (a; q), β (a; q)) given by (54a) α L(a; q) ( ) L b L q L (aq/b; q ) L (bq; q ) L α L (a; q ), α L+(a; q) 0, (54b) β L(a; (b; q ) L q) (q, b; q) L (aq; q ) L L/ (aq/b; q ) (q L ; q) q (q L /b; q ) β (a; q ) Fo b 0, b, b a / and b a / q this coesponds to the even j case of (3), (5), (8) and (9) Poof Witing the nontivial pat of (54) as α L(a; q) h L (a, b)α L (a; q ), β L(a; q) L/ f L, (a, b)β (a; q ), the claim of the lemma boils down to showing that L/ s f L, (a, b) (q ; q ) s (aq ; q ) +s h s (a, b) (q; q) L s (aq; q) L+s Afte shifting + s this follows fom (0) with (a, b, n) (aq 4s+, bq s, L s) Next, () and (3) fo even j coespond to the following two quatic Bailey lemmas Lemma 55 If (α(a; q), β(a; q)) is a Bailey pai elative to a and q, then so is (α (a; q), β (a; q)) given by (55a) (55b) α L(a; q) α L (a ; q 4 ), α L+(a; q) 0, β L(a; q) ql ( q ; q ) L (q, aq; q ) L L/ ( aq, q L ; q ) q 4 ( q 3 L ; q ) β (a ; q 4 ) Poof Copying the poof of Lemma 54 this follows fom (4) with (a, n) ( aq 4s+, L s)