Some new results on Andrews' and Warnaar's q-identities

Some new results on Andrews’ and W arnaar’s q -iden tities Qi Chen a,1 a Dep artment of Mathematics, So o chow University, SuZhou 215006, P.R.China Abstract In this pap er, by the technique of inv erse relations and comparing coefficients, we establish some generalized forms of Andrews’ q -series iden tity and t wo new Bailey pairs and q - iden tities closely related to Andrews-W arnaar’s sum iden tity for partial theta functions. Keywor ds: q -Series, In v erse relation, Comparing co efficien t, P artial theta function, Andrews iden tity , W arnaar identit y . AMS sub ject classification (2020) : Primary 05A30; Secondary 33D15. 1. In tro duction In the theory of q -series, there is a w ell-kno wn identit y given b y G.E. Andrews. Theorem 1.1 (Andrews’ iden tity: [ 4 , (I I. 11)]) . ∞ X n =0 ( a, b ; q ) n ( q ; q ) n ( abq ; q 2 ) n q n ( n +1) 2 = ( − q ; q ) ∞ ( aq , bq ; q 2 ) ∞ ( abq ; q 2 ) ∞ . A quic k glance on this identit y sho ws that the series expansion of the factors ( ax ; q 2 ) ∞ ( x ; q ) ∞ and ( ax ; q ) ∞ ( x ; q 2 ) ∞ (1.1) seem necessary but to ha v e been ignored, merely comparing with the q -binomial theorem [ 4 , (I I. 3)] ( ax ; q ) ∞ ( x ; q ) ∞ = ∞ X n =0 ( a ; q ) n ( q ; q ) n x n . (1.2) The latter has b een frequently used in study of q -series. Therefore, it is natural to consider the former’s expansion as p o wer series of x . W e believe such expansions also pla y a similar role as the q -binomial theorem in the study of q -series. As an initial p ositiv e answer to this question, in this pap er we will establish the follo wing q -identities, each of whic h can b e regarded as generalizations of Theorem 1.1 . 1 E-mail address: qc hen2025@stu.suda.edu.cn Theorem 1.2 (Generalization of Andrews’ identit y-I) . F or any c omplex numb ers a, b, and c , ther e holds ∞ X n =0 ( a, b ; q ) n ( q ; q ) n ( abq ; q 2 ) n q n ( n +1) 2 c n = ( − cq , a, b ; q ) ∞ ( abq ; q 2 ) ∞ ∞ X n =0 h n ( a, b | q 2 ) ( q , − cq ; q ) n , (1.3) wher e the R o gers-Sze gö p olynomials ar e define d by h n ( a, b | q ) := n X k =0  n k  q a k b n − k . (1.4) Theorem 1.3 (Generalization of Andrews’ iden tity-II) . L et h n ( a, b | q ) b e define d by ( 1.4 ) . W e have ∞ X n =0 ( a, b ; q ) n ( q ; q ) n ( ab ; q 2 ) n q n ( n +1) 2 c n = ( a, b, − cq ; q ) ∞ ( ab ; q 2 ) ∞ ∞ X n =0 h n ( a, bq | q 2 ) + h n ( aq , b | q 2 ) ( q , − cq ; q ) n (1 + q n ) . Similar to but sligh tly different from these tw o conclusions is the follo wing q -identit y . Theorem 1.4. ∞ X n =0 ( a, b ; q ) n ( q , ( ab ) 1 / 2 ; q ) n ( − ( ab ) 1 / 2 ; q ) n +1 q n ( n +1) 2 c n (1.5) = ( a, b, − cq ; q ) ∞ ( ab ; q 2 ) ∞ ∞ X n =0  a 1 / 2 a 1 / 2 + b 1 / 2 h n ( a, bq | q 2 ) ( q , − cq ; q ) n + b 1 / 2 a 1 / 2 + b 1 / 2 h n ( aq , b | q 2 ) ( q , − cq ; q ) n  . It is w orth men tioning that the case c = 1 of Theorem 1.4 is v ery similar to [ 5 , Thm. 1], the latter is used to generating functions for the partitions of asymmetric residue classes. The reader is referred to lo c. cit for more details. Here and in what follo ws, we will adopt the standard notation and terminology for basic h yp ergeometric series in the b o ok [ 4 ]. Let | q | < 1 and the q -shifted factorials b e given b y ( x ; q ) ∞ = ∞ Y n =0 (1 − xq n ) , ( x ; q ) n = ( x ; q ) ∞ ( xq n ; q ) ∞ . F or conv enience, w e will adopt the following notation for multiple q -shifted factorial ( x 1 , x 2 , · · · , x m ; q ) n = ( x 1 ; q ) n ( x 2 ; q ) n · · · ( x m ; q ) n . The basic h yp ergeometric series with the base q and the argumen t x is defined by r ϕ s  a 1 , a 2 , · · · , a r b 1 , b 2 , · · · , b s ; q , x  = ∞ X n =0 ( a 1 , a 2 , · · · , a r ; q ) n ( q , b 1 , b 2 , · · · , b s ; q ) n ( τ 1 ( n )) s +1 − r x n , (1.6) where τ r ( n ) = ( − 1) n q rn ( n − 1) / 2 , { b i } are complex num b ers such that none of the denomina- tors in ( 1.6 ) v anishes, whic h we never mention as conditions in what follows unless otherwise stated. As a custom, the q -binomial co efficient  n k  q is defined b y  n k  q := ( q ; q ) n ( q ; q ) k ( q ; q ) n − k . 2 Our pap er is organized as follo ws. In the next section we will sho w some preliminaries and then give the full pro ofs for Theorems 1.2 , 1.3 , and 1.4 . After that, some concrete q -identities will b e discussed. In Section 3 , we inv estigate W arnaar’s partial theta identities in the same v ein, tw o new Bailey pairs and some q -iden tities are presented. 2. Pro ofs of lemmas and theorems 2.1. Pr eliminary lemmas As a p ositive answer to the expansion question of ( 1.1 ) and one of the necessary pre- liminary results, w e now show Lemma 2.1. Supp ose that ( ax ; q 2 ) ∞ ( x ; q ) ∞ = ∞ X n =0 λ n ( a ) x n . (2.1) Then the c o efficients λ n ( a ) = 2 ϕ 1  q − n , − q − n 0 ; q , q 2 /a  ( − a ) n q n ( n − 1) ( q 2 ; q 2 ) n . (2.2) Pro of. It is obvious that ( 2.1 ) is equiv alen t to 1 ( x ; q ) ∞ = 1 ( ax ; q 2 ) ∞ ∞ X n =0 λ n ( a ) x n . Namely ∞ X n =0 x n ( q ; q ) n = ∞ X k =0 a k x k ( q 2 ; q 2 ) k ∞ X n =0 λ n ( a ) x n . By equating the co efficients of x n , w e get 1 ( q ; q ) n = n X k =0 a n − k ( q 2 ; q 2 ) n − k λ k ( a ) , whic h is ( q 2 ; q 2 ) n ( q ; q ) n a − n = n X k =0  n k  q 2 ( q 2 ; q 2 ) k a − k λ k ( a ) . After simplification, it tak es the form α n = n X k =0  n k  q 2 β k , (2.3) where α n = ( − q ; q ) n /a n , β n = ( q 2 ; q 2 ) n λ n ( a ) /a n . 3 Recall that there is a matrix in version  n k  q 2 ! − 1 n ≥ k ≥ 0 =  n k  q 2 τ 2 ( n − k ) ! n ≥ k ≥ 0 . By applying this in verse relation, we solv e λ n ( a ) from ( 2.3 ) as b elow ( q 2 ; q 2 ) n λ n ( a ) a − n = n X k =0  n k  q 2 τ 2 ( n − k )( − q ; q ) k a − k = τ 2 ( n ) n X k =0  n k  q 2 τ 2 ( k ) q 2( k − nk ) ( − q ; q ) k a − k . A bit simplification giv es λ n ( a ) = a n τ 2 ( n ) ( q 2 ; q 2 ) n n X k =0 ( q − 2 n ; q 2 ) k ( q 2 ; q 2 ) k ( − q ; q ) k ( q 2 /a ) k = a n τ 2 ( n ) ( q 2 ; q 2 ) n 2 ϕ 1  q − n , − q − n 0 ; q , q 2 /a  . That is w e wan ted. Apart from Lemma 2.1 , we also require the follo wing recurrence relation related to λ n ( a ) . Lemma 2.2. Define, for inte gers r, s and n , the finite sum T r,n ( s ) := n X k =0 ( q − 2 n ; q 2 ) k ( q , q 1+ r − n ; q ) k q (2 − s ) k . (2.4) Then ther e is a r e curr enc e r elation of de gr e e two for { T r,n ( s ) } n ≥ 0 q − s (1 − q 2 n +2 )( q − s + q r + n ) T r,n ( s ) + q n ( q n − q r )( − q 2 n +1 + q r + n + q − s + q − s − 1 ) T r,n +1 ( s ) + q 2 n ( q n − q r )( q n +1 − q r ) T r,n +2 ( s ) = 0 . (2.5) Pro of. F or con v enience, we write y for q s . Then the recurrence relation ( 2.5 ) b ecomes y − 1  1 − q 2 n +2   y − 1 + q r + n  n X k =0 ( q − 2 n ; q 2 ) k ( q , q 1+ r − n ; q ) k q 2 k y − k + q n ( q n − q r )  − q 1+2 n + q r + n + (1 + q ) q − 1 y − 1  n +1 X k =0 ( q − 2 n − 2 ; q 2 ) k ( q , q r − n ; q ) k q 2 k y − k = − q 2 n ( q n − q r )  q n +1 − q r  n +2 X k =0 ( q − 2 n − 4 ; q 2 ) k ( q , q r − n − 1 ; q ) k q 2 k y − k . (2.6) As such, in order to show ( 2.6 ) w e only need to compare the co efficients of y − k on b oth sides. Th us, w e hav e [ y − k ] LHS of ( 2.6 ) =  1 − q 2 n +2   q − 2 n ; q 2  k − 2 ( q , q 1+ r − n ; q ) k − 2 q 2( k − 2) (= S 1 ) 4 +  1 − q 2 n +2  q r + n  q − 2 n ; q 2  k − 1 ( q , q 1+ r − n ; q 2 ) k − 1 q 2( k − 1) (= S 2 ) + q n ( q n − q r )  − q 1+2 n + q r + n   q − 2 n − 2 ; q 2  k ( q , q r − n ; q ) k q 2 k (= S 3 ) + q n ( q n − q r ) (1 + q ) q − 1  q − 2 n − 2 ; q 2  k − 1 ( q , q r − n ; q ) k − 1 q 2( k − 1) (= S 4 ) . Here and in the sequel, the notation [ t n ] f ( t ) denotes the co efficient of t n in f ( t ) . Conse- quen tly , it is easy to c heck S 1 + S 4 = ( q − 2 n − 4 ; q 2 ) k ( q , q r − n − 1 ; q ) k q 2 n +2 k − 3 (1 − q r − n − 1 )(1 − q r − n )(1 − q 2 k ) 1 − q − 2 n − 4 , S 2 + S 3 = −  q − 2 n − 2 ; q 2  k ( q , q r − n ; q ) k q 2 k +4 n +1 (1 − q r − n )(1 − q r − n + k − 1 ) . Based on these t wo results, it follows [ y − k ] LHS of ( 2.6 ) = ( q − 2 n − 4 ; q 2 ) k ( q , q r − n − 1 ; q ) k q 2 k  q 2 n − 3 (1 − q r − n − 1 )(1 − q r − n )(1 − q 2 k ) 1 − q − 2 n − 4 − q 4 n +1 (1 − q r − n − 1 )(1 − q − 2 n − 4+2 k ) 1 − q − 2 n − 4 (1 − q r − n )  = ( q − 2 n − 4 ; q 2 ) k ( q , q r − n − 1 ; q ) k q 2 k ( − q 4 n +1 )(1 − q − 2 n − 4 ) (1 − q r − n − 1 )(1 − q r − n ) 1 − q − 2 n − 4 = [ y − k ] RHS of ( 2.6 ) . As desired. V ery interesting is that ( 2.5 ) includes the q -Chu-V andermonde formula [ 4 , (I I. 6)] as a sp ecial case. Lemma 2.3. L et T r,n ( s ) b e given by ( 2.4 ) . Then the fol lowing identities hold. T r,n (1) = ( − q 1+ r ; q ) n ( q 1+ r − n ; q ) n ( − 1) n q − n 2 , (2.7) T r,n (0) = q n + q r 1 + q r + n ( − q 1+ r ; q ) n ( q 1+ r − n ; q ) n ( − 1) n q − n 2 + n . (2.8) 2.2. The ful l pr o ofs of the main the or ems As is exp ected, we turn to present full pro ofs for Theorems 1.2 - 1.4 mainly based on Lemmas 2.1 - 2.3 . As the first step, w e need to sho w a master theorem. Theorem 2.4. L et T r,n ( s ) b e given by ( 2.4 ) . Then ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( abq s ; q 2 ) n = ( a, b, − cq ; q ) ∞ ( abq s ; q 2 ) ∞ X n ≥ k ≥ 0 τ 2 ( k )( bq s ) k ( q 2 ; q 2 ) k a n − k ( q ; q ) n − 2 k T n − k,k ( s ) ( − cq ; q ) n . (2.9) 5 Pro of. It is clear that from Lemma 2.1 , it follows ( abq s +2 n ; q 2 ) ∞ ( bq n ; q ) ∞ = ∞ X M =0 ( aq s + n ) M τ 2 ( M ) ( q 2 ; q 2 ) M 2 ϕ 1  q − M , − q − M 0 ; q , q 2 − s − n /a  ( bq n ) M . Th us ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( abq s ; q 2 ) n = ( b ; q ) ∞ ( abq s ; q 2 ) ∞ ∞ X n =0 ( a ; q ) n ( q ; q ) n q n ( n +1) / 2 c n × ∞ X M =0 ( aq s + n ) M τ 2 ( M ) ( q 2 ; q 2 ) M 2 ϕ 1  q − M , − q − M 0 ; q , q 2 − s − n /a  ( bq n ) M . W ritten out in explicit terms, it is ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( abq s ; q 2 ) n = ( a, b ; q ) ∞ ( abq s ; q 2 ) ∞ ∞ X n =0 q n ( n +1) / 2 c n ( q ; q ) n ∞ X M =0 q ( s + n ) M τ 2 ( M ) ( q 2 ; q 2 ) M × M X k =0 ( q − 2 M ; q 2 ) k ( q ; q ) k q k (2 − s − n ) ( bq n ) M a M − k ( aq n ; q ) ∞ . Rearrange the righ t-hand infinite series as p ow er series in a . W e hav e ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( abq s ; q 2 ) n = ( a, b ; q ) ∞ ( abq s ; q 2 ) ∞ ∞ X n =0 q n ( n +1) / 2 c n ( q ; q ) n ∞ X M =0 q ( s + n ) M τ 2 ( M ) ( q 2 ; q 2 ) M × M X k =0 ( q − 2 M ; q 2 ) k ( q ; q ) k q k (2 − s − n ) ( bq n ) M ∞ X r =0 q n ( r − ( M − k )) ( q ; q ) r − ( M − k ) a r . After simplification and rearrangemen t of summations ov er n and k , it b ecomes ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( abq s ; q 2 ) n = ( a, b ; q ) ∞ ( abq s ; q 2 ) ∞ ∞ X M =0 ( bq s ) M τ 2 ( M ) ( q 2 ; q 2 ) M ∞ X r =0 a r ( q ; q ) r − M × M X k =0 ( q − 2 M ; q 2 ) k ( q , q 1+ r − M ; q ) k q k (2 − s ) ∞ X n =0 q n ( n − 1) / 2 c n ( q ; q ) n q n (1+ r + M ) . Observ e that the last summation can b e ev aluated in closed form by the q -binomial theorem ( 1.2 ). Th us w e get ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( abq s ; q 2 ) n = ( a, b, − cq ; q ) ∞ ( abq s ; q 2 ) ∞ ∞ X M =0 ( bq s ) M τ 2 ( M ) ( q 2 ; q 2 ) M ∞ X r =0 a r ( q ; q ) r − M ( − cq ; q ) r + M M X k =0 ( q − 2 M ; q 2 ) k q k (2 − s ) ( q , q 1+ r − M ; q ) k N := r + M === ( a, b, − cq ; q ) ∞ ( abq s ; q 2 ) ∞ ∞ X N =0 1 ( − cq ; q ) N ∞ X M =0 ( bq s ) M τ 2 ( M ) a N − M ( q 2 ; q 2 ) M ( q ; q ) N − 2 M M X k =0 ( q − 2 M ; q 2 ) k q k (2 − s ) ( q , q 1+ N − 2 M ; q ) k . Reform ulating this identit y in terms of ( 2.4 ), we get ( 2.9 ). First, b y utilizing Theorem 2.4 and the iden tit y ( 2.7 ), we begin to show Theorem 1.2 . 6 Pro of of Theorem 1.2 . Observ e that the sp ecial case s = 1 of ( 2.9 ) yields ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( abq ; q 2 ) n = ( a, b, − cq ; q ) ∞ ( abq ; q 2 ) ∞ X n ≥ k ≥ 0 τ 2 ( k )( bq ) k ( q 2 ; q 2 ) k a n − k ( q ; q ) n − 2 k T n − k,k (1) ( − cq ; q ) n . Substituting ( 2.7 ) into T n − k,k (1) , w e can derive ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( abq ; q 2 ) n = ( a, b, − cq ; q ) ∞ ( abq ; q 2 ) ∞ ∞ X n =0 1 ( − cq ; q ) n × n X k =0 τ 2 ( k )( bq ) k ( q 2 ; q 2 ) k a n − k ( q ; q ) n − 2 k ( − q 1+ n − k ; q ) k ( q 1+ n − 2 k ; q ) k ( − q − k ) k = ( a, b, − cq ; q ) ∞ ( abq ; q 2 ) ∞ ∞ X n =0 ( − q ; q ) n ( − cq ; q ) n n X k =0 b k a n − k ( q 2 ; q 2 ) k ( q 2 ; q 2 ) n − k = ( a, b, − cq ; q ) ∞ ( abq ; q 2 ) ∞ ∞ X n =0 1 ( q , − cq ; q ) n n X k =0  n k  q 2 b k a n − k . Note that the inner sum is the Rogers-Szegö p olynomials ( 1.4 ) on the base q 2 . The pro of is finished. F rom ( 1.3 ) of Theorem 1.2 , we can deduce some interesting q -iden tities. F or instance, letting c = 0 in ( 1.3 ), we hav e a new generating function of the Rogers-Szegö p olynomials. Corollary 2.5. L et h n ( a, b | q ) b e define d by ( 1.4 ) . Then we have ∞ X n =0 h n ( a, b | q 2 ) ( q ; q ) n = ( abq ; q 2 ) ∞ ( a, b ; q ) ∞ . (2.10) A com bination of this result with Lemma 2.1 gives rise to a finite q -series identit y ( − a ) M q M 2 M X k =0 ( q − 2 M ; q 2 ) k ( q ; q ) k ( q /a ) k = M X k =0  M k  q ( a ; q ) k q k ( k +1) / 2 , whic h is in agreement with the sp ecial case ( b, c, z ) = ( a, 0 , − q M +1 ) of [ 4 , (I I I. 6)]. Alternativ ely , let c = q − 1 in ( 1.3 ). Then w e obtain Corollary 2.6. ∞ X n =0 ( a, b ; q ) n ( q ; q ) n ( abq ; q 2 ) n q n ( n − 1) 2 = ( a, b ; q 2 ) ∞ + ( aq , bq ; q 2 ) ∞ ( q , abq ; q 2 ) ∞ . (2.11) Pro of. It is clear that when c = q − 1 , Theorem 1.2 reduces to ∞ X n =0 ( a, b ; q ) n q n ( n − 1) 2 ( q ; q ) n ( abq ; q 2 ) n = ( − 1 , a, b ; q ) ∞ ( abq ; q 2 ) ∞ ∞ X n =0 b n ( q , − 1; q 2 ) n n X k =0  n k  q 2 ( a/b ) k 7 = ( − q , a, b ; q ) ∞ ( abq ; q 2 ) ∞ ∞ X n =0 b n ( q 2 ; q 2 ) n (1 + q n ) n X k =0  n k  q 2 ( a/b ) k = ( − q , a, b ; q ) ∞ ( abq ; q 2 ) ∞  S ( a, b ) + S ( aq , bq )  , (2.12) where S ( a, b ) := ∞ X n =0 b n ( q 2 ; q 2 ) n n X k =0  n k  q 2 ( a/b ) k . It is easy to see that S ( a, b ) = 1 ( a, b ; q 2 ) ∞ , ( − q ; q ) ∞ = 1 ( q ; q 2 ) ∞ . A substitution of these t wo results into ( 2.12 ) giv es rise to ( 2.11 ). With the help of ( 2.8 ), w e ma y sho w Theorem 1.3 directly . Pro of of Theorem 1.3 . Let s = 0 in Theorem 2.4 and we appeal to Lemma 2.3 to ev al- uate T n − k,k (0) . The result is as follo ws. ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( ab ; q 2 ) n = ( a, b, − cq ; q ) ∞ ( ab ; q 2 ) ∞ ∞ X n =0 1 ( − cq ; q ) n × n X k =0 τ 2 ( k ) b k ( q 2 ; q 2 ) k a n − k ( q ; q ) n − 2 k q k + q n − k 1 + q n ( − q 1+ n − k ; q ) k ( q 1+ n − 2 k ; q ) k ( − 1) k q − k 2 + k = ( a, b, − cq ; q ) ∞ ( ab ; q 2 ) ∞ ∞ X n =0 ( − q ; q ) n ( − cq ; q ) n n X k =0 b k ( q 2 ; q 2 ) k a n − k ( q 2 ; q 2 ) n − k q k + q n − k 1 + q n . F urther simplification leads to ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( ab ; q 2 ) n = ( a, b, − cq ; q ) ∞ ( ab ; q 2 ) ∞ ∞ X n =0 1 ( q , − cq ; q ) n n X k =0  n k  q 2 b k a n − k q k + q n − k 1 + q n = ( a, b, − cq ; q ) ∞ ( ab ; q 2 ) ∞ ∞ X n =0 1 ( q , − cq ; q ) n (1 + q n ) × n X k =0  n k  q 2 ( bq ) k a n − k + n X k =0  n k  q 2 b k ( aq ) n − k ! . Reform ulate in terms of the Rogers-Szegö p olynomials ( 1.4 ), this yields the desired result. The theorem is pro ved. As a last application of Theorem 2.4 , w e can show Theorem 1.4 . Pro of of Theorem 1.4 . T o show ( 1.5 ), we first split it into t w o parts, as b elow: ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q , ( ab ) 1 / 2 , − q ( ab ) 1 / 2 ; q ) n = ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( abq 2 ; q 2 ) n (1 − bq n ) + ( b/a ) 1 / 2 (1 − aq n ) (1 + ( b/a ) 1 / 2 )(1 − ( ab ) 1 / 2 ) 8 = (1 − b ) L 1 + (1 − a )( b/a ) 1 / 2 L 2 (1 + ( b/a ) 1 / 2 )(1 − ( ab ) 1 / 2 ) . Here, w e define L 1 := ∞ X n =0 ( a, bq ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( abq 2 ; q 2 ) n , L 2 := ∞ X n =0 ( aq , b ; q ) n q n ( n +1) 2 c n ( q ; q ) n ( abq 2 ; q 2 ) n . F rom Theorem 1.2 it follows, respectively , L 1 = ( − cq , a, bq ; q ) ∞ ( abq 2 ; q 2 ) ∞ ∞ X n =0 h n ( a, bq | q 2 ) ( q , − cq ; q ) n , L 2 = ( − cq , aq , b ; q ) ∞ ( abq 2 ; q 2 ) ∞ ∞ X n =0 h n ( aq , b | q 2 ) ( q , − cq ; q ) n . After a bit simplification, w e get ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 c n ( q , ( ab ) 1 / 2 , − q ( ab ) 1 / 2 ; q ) n = 1 (1 + ( b/a ) 1 / 2 )(1 − ( ab ) 1 / 2 ) ( − cq , a, b ; q ) ∞ ( abq 2 ; q 2 ) ∞ × ∞ X n =0 h n ( a, bq | q 2 ) ( q , − cq ; q ) n + ( b/a ) 1 / 2 ∞ X n =0 h n ( aq , b | q 2 ) ( q , − cq ; q ) n ! . Dividing b oth sides by 1 + ( ab ) 1 / 2 leads us to the desired iden tity . The case c = 1 of Theorem 1.4 yields the following result whic h is very similar to [ 5 , Thm. 1]. Corollary 2.7. ∞ X n =0 ( a, b ; q ) n q n ( n +1) 2 ( q , ( ab ) 1 / 2 ; q ) n ( − ( ab ) 1 / 2 ; q ) n +1 = a 1 / 2 ( aq , b ; q 2 ) ∞ + b 1 / 2 ( a, bq ; q 2 ) ∞ ( a 1 / 2 + b 1 / 2 )( q , ab ; q 2 ) ∞ . (2.13) Pro of. When c = 1 , the conclusion follo ws from ( 1.5 ) directly by using ( 2.10 ). 3. New pro of of W arnaar’s partial theta identit y Recall that in their pap er [ 3 ], Andrews and W arnaar established the following b eautiful partial theta iden tity . Lemma 3.1 (Cf. [ 7 , Thm. 1.5]) . 1 + ∞ X n =1 ( − 1) n q ( n 2 ) ( a n + b n ) = ( q , a, b ; q ) ∞ ∞ X n =0 ( ab/q ; q ) 2 n ( q , a, b, ab ; q ) n q n . (3.1) It is w orth mentioning that in their pap er [ 6 ], W ang and Ma put forward a unified metho d to such kind of theta iden tities. In this part, we will fo cus on the sp ecial case of ( 3.1 ) and treat it via the foregoing argumen t. 9 Theorem 3.2 (W arnaar’s partial theta identit y: [ 7 , p. 4]) . 1 + 2 ∞ X n =1 a n q 2 n 2 = ( q ; q ) ∞ ( aq ; q 2 ) ∞ ∞ X n =0 ( − a ; q ) 2 n q n ( q , − aq ; q ) n ( aq ; q 2 ) n . (3.2) During this pro cedure, we come to a new Bailey pair and some allied q -identities ly- ing behind this theta identit y . T o this end, we recall first the well known Bailey lemma asso ciated with the Bailey pairs. Lemma 3.3 (Bailey lemma: [ 2 , Chap. 3]) . F or any inte ger n ≥ 0 , it holds 1 ( aq /ρ 1 , aq /ρ 2 ; q ) n n X k =0 ( ρ 1 , ρ 2 ; q ) k ( aq /ρ 1 ρ 2 ; q ) n − k ( q ; q ) n − k  aq ρ 1 ρ 2  k β k ( a, q ) = n X k =0 ( ρ 1 , ρ 2 ; q ) k ( q ; q ) n − k ( aq ; q ) n + k ( aq /ρ 1 , aq /ρ 2 ; q ) k  aq ρ 1 ρ 2  k α k ( a, q ) , (3.3) wher e ( α n ( a, q ) , β n ( a, q )) is a Bailey p air with r esp e ct to a and q , define d by β n ( a, q ) = n X k =0 α k ( a, q ) ( q ; q ) n − k ( aq ; q ) n + k . (3.4) Using this lemma, w e can show Prop osition 3.4. Define, for any inte ger n ≥ 0 , the finite sum γ ( n ) := n X k =0  n k  q q k 2 ( − q ; q ) k . (3.5) Then ( α n (1 , q ) , β n (1 , q )) =  2( − 1) n q 2 n 2 , 1 ( q ; q ) 2 n + γ ( n ) ( q ; q ) n  (3.6) is a Bailey p air with r esp e ct to a = 1 . Pro of. First of all, letting ρ 1 , ρ 2 → ∞ and a = 1 , then ( 3.3 ) reduces to n X k =0 q k 2 ( q ; q ) n − k β k (1 , q ) = n X k =0 q k 2 ( q ; q ) n + k ( q ; q ) n − k α k (1 , q ) . (3.7) In suc h case, we see that ( 3.4 ) reduces to β n (1 , q ) = n X k =0 α k (1 , q ) ( q ; q ) n − k ( q ; q ) n + k . (3.8) In order to sho w that ( 3.6 ) is a Bailey pair, we first set β n (1 , q ) := 1 ( q 2 ; q 2 ) n + 1 ( q ; q ) 2 n . 10 By the in verse relation, we ma y solv e α n (1 , q ) from ( 3.8 ), as follows α n (1 , q ) (1 + q n ) τ ( n ) = n X k =0 ( q − n , q n ; q ) k ( q ; q ) 2 k q k + n X k =0 ( q − n , q n ; q ) k ( q 2 ; q 2 ) k q k , whic h, after an application of the q -Chu-V andermonde formula ( 2.7 ), turns out to be α n (1; q ) = (1 + q n ) τ ( n ) ( − q 1 − n ; q ) n ( − q ; q ) n q n 2 = 2( − 1) n q n 2 . Therefore, w e come to a Bailey pair  2( − 1) n q n 2 , 1 ( q 2 ; q 2 ) n + 1 ( q ; q ) 2 n  . (3.9) A direct substitution of ( 3.9 ) in to ( 3.7 ) giv es rise to ( 3.6 ). With the ab ov e setup, w e are no w in go o d position to show W arnaar’s theta identit y ( 3.1 ). Pro of of Theorem 3.2 . It suffices to ev aluate the sum on the righ t-hand side of ( 3.2 ). A t first, we see RHS of ( 3.2 ) = ( q ; q ) ∞  aq ; q 2  ∞ ∞ X n =0 ( − a ; q ) 2 n q n ( q , − aq ; q ) n ( aq ; q 2 ) n = ( q , − a ; q ) ∞ ( − aq ; q ) ∞ ∞ X n =0 q n ( q ; q ) n ( aq 1+2 n ; q 2 ) ∞ ( − aq 2 n ; q ) ∞ ( − aq 1+ n ; q ) ∞ . By Lemma 2.1 , w e hav e RHS of ( 3.2 ) = ( q ; q ) ∞ (1 + a ) ∞ X n,M =0 q n ( q ; q ) n X i + j = M q (1+ n ) i q ( i 2 ) ( q ; q ) i λ j ( − q )( − q 2 n ) j a M = ( q ; q ) ∞ (1 + a ) X M ≥ j ≥ 0 q ( M − j 2 ) q M − j ( q ; q ) M − j ( − 1) j λ j ( − q ) a M ∞ X n =0 q (1+ M + j ) n ( q ; q ) n = (1 + a ) X M ≥ j ≥ 0 ( q ; q ) M + j ( q ; q ) M − j ( − 1) j q ( M − j 2 ) q M − j λ j ( − q ) a M , where λ j ( − q ) is given b y ( 2.2 ). Relab eling M b y n and j by k , we get RHS of ( 3.2 ) = (1 + a ) ∞ X n =0 a n q n + ( n 2 ) n X k =0 ( q ; q ) n + k ( q ; q ) n − k ( − 1) k q ( k 2 ) − nk λ k ( − q ) = ∞ X n =0 a n F ( n ) + ∞ X n =0 a n +1 F ( n ) = 1 + ∞ X n =1 a n ( F ( n ) + F ( n − 1)) , where F ( n ) := q n + ( n 2 ) n X k =0 ( q ; q ) n + k ( q ; q ) n − k ( − 1) k q ( k 2 ) − nk λ k ( − q ) . 11 It is not hard to c heck that F ( n ) + F ( n − 1) = q ( n 2 ) n X k =0 ( q ; q ) n + k − 1 ( q ; q ) n − k τ ( k ) q − nk λ k ( − q )  q n (1 − q n + k ) + q k (1 − q n − k )  = q ( n 2 ) (1 − q 2 n ) n X k =0 ( q ; q ) n + k − 1 ( q ; q ) n − k τ ( k ) q k − nk λ k ( − q ) = q ( n 2 ) (1 + q n ) n X k =0 ( q − n , q n ; q ) k q k λ k ( − q ) . On comparing with the left-hand side of ( 3.2 ), clearly w e only need to sho w q ( n 2 ) (1 + q n ) n X k =0 ( q − n , q n ; q ) k q k λ k ( − q ) = 2 q 2 n 2 . (3.10) The argumen t go es as follows. First, we recast ( 3.10 ) as the form β n = n X k =0 ( q − n , q n ; q ) k q k λ k ( − q ) , (3.11) where          λ n ( − q ) = q n 2 ( q 2 ; q 2 ) n 2 ϕ 1  q − n , − q − n 0 ; q , − q  β n = 2 q 2 n 2 − ( n 2 ) 1 + q n ( n ≥ 1); β 0 = 1 . By virtue of Carlitz’s matrix in version  ( q − n , aq n ; q ) k ( q , aq ; q ) k q k  − 1 n ≥ k ≥ 0 =  ( a, q − n ; q ) k ( q , aq 1+ n ; q ) k 1 − aq 2 k 1 − a q kn  n ≥ k ≥ 0 , it is easily seen that ( 3.11 ) is equiv alent to ( q ; q ) 2 n λ n ( − q ) = 1 + n X k =1 ( q − n ; q ) k ( q ; q ) k − 1 ( q , q 1+ n ; q ) k (1 − q 2 k ) q kn β k . That is λ n ( − q ) = 1 ( q ; q ) 2 n + 1 ( q ; q ) n n X k =1 ( q − n ; q ) k ( q ; q ) k − 1 ( q ; q ) k ( q ; q ) n + k (1 − q 2 k ) q kn β k = 1 ( q ; q ) 2 n + n X k =1 τ ( k )(1 + q k ) ( q ; q ) n − k ( q ; q ) n + k β k = n X k = − n ( − 1) k q 2 k 2 ( q ; q ) n − k ( q ; q ) n + k . Lastly , we get q n 2 ( q 2 ; q 2 ) n n X k =0 ( q − n , − q − n ; q ) k ( q ; q ) k ( − q ) k = n X k = − n ( − 1) k q 2 k 2 ( q ; q ) n − k ( q ; q ) n + k . Using the basic relation [ 4 , (I. 10)] ( − q − n ; q ) k = ( − q ; q ) n ( − q ; q ) n − k ( − 1) k q − nk τ ( k ) to simplify the last iden tity , only finding that it is the Bailey lemma given b y the Bailey pair ( 3.9 ). Summing up, ( 3.10 ) is pro v ed. 12 W e end our discussion by the follo wing q -identities via com bination of tw o Bailey pairs ( 3.6 ) and ( 3.9 ) with the Bailey lemma. Prop osition 3.5. With the same notation as L emma 3.3 and γ ( n ) b e given by ( 3.5 ) . Then we have 1 ( q /ρ 1 , q /ρ 2 ; q ) n n X k =0 ( ρ 1 , ρ 2 ; q ) k ( q /ρ 1 ρ 2 ; q ) n − k ( q ; q ) k ( q ; q ) n − k  q ρ 1 ρ 2  k  1 ( q ; q ) k + γ ( k )  = 2 n X k =0 ( ρ 1 , ρ 2 ; q ) k q 2 k 2 ( q ; q ) n − k ( q ; q ) n + k ( q /ρ 1 , q /ρ 2 ; q ) k  − q ρ 1 ρ 2  k . (3.12) In p articular (i) ( ρ 1 = 1 /a, ρ 2 = q in ( 3.12 )) F or | a | < 1 , it holds n X k =0 (1 /a ; q ) k ( a ; q ) n − k ( q ; q ) n − k a k γ ( k ) = 2( q ; q ) n − 1 ( aq ; q ) n n X k =0 (1 /a ; q ) k (1 − q k ) q 2 k 2 ( aq ; q ) k ( q ; q ) n − k ( q ; q ) n + k ( − a ) k . The limitation n → ∞ yields (1 − a ) ∞ X k =0 (1 /a ; q ) k a k γ ( k ) = 2 ∞ X k =0 (1 /a ; q ) k ( aq ; q ) k ( − a ) k (1 − q k ) q 2 k 2 . (ii) ( ρ 1 , ρ 2 → ∞ in ( 3.12 )) n X k =0 q k 2 γ ( k ) ( q ; q ) k ( q ; q ) n − k = n X k = − n ( − 1) k q 3 k 2 ( q ; q ) n − k ( q ; q ) n + k , whose limitation n → ∞ yields ∞ X k =0 q k 2 γ ( k ) ( q ; q ) k = ( q 3 ; q 6 ) ∞ ( q , q 2 ; q 3 ) ∞ . (3.13) Pro of. Actually , ( 3.12 ) is the direct consequence of the Bailey lemma sp ecialized by the Bailey pair ( 3.6 ), while ( 3.13 ) in volv es the Jacobi triple product identit y ∞ X k = −∞ ( − 1) k q 3 k 2 = ( q 3 , q 3 , q 6 ; q 6 ) ∞ . When the Bailey pair ( 3.9 ) is tak en into account, w e also ha ve Prop osition 3.6. With the same notation as L emma 3.3 . W e have 1 ( q /ρ 1 , q /ρ 2 ; q ) n n X k =0 ( ρ 1 , ρ 2 ; q ) k ( q /ρ 1 ρ 2 ; q ) n − k ( q ; q ) n − k  q ρ 1 ρ 2  k ( q ; q ) k + ( − q ; q ) k ( q 2 ; q 2 ) k ( q ; q ) k 13 = 2 n X k =0 ( ρ 1 , ρ 2 ; q ) k q k 2 ( q ; q ) n − k ( q ; q ) n + k ( q /ρ 1 , q /ρ 2 ; q ) k  − q ρ 1 ρ 2  k . (3.14) In p articular (i) ( ρ 1 = 1 /a, ρ 2 = q in ( 3.14 )) F or | a | < 1 , it holds n X k =0 (1 /a ; q ) k ( a ; q ) n − k ( q ; q ) n − k ( − q ; q ) k a k = 2( q ; q ) n − 1 ( aq ; q ) n n X k =0 (1 /a ; q ) k (1 − q k ) q k 2 ( aq ; q ) k ( q ; q ) n − k ( q ; q ) n + k ( − a ) k . (ii) ( ρ 1 , ρ 2 → ∞ in ( 3.14 )) n X k =0 q k 2 ( q 2 ; q 2 ) k ( q ; q ) n − k = n X k = − n ( − 1) k q 2 k 2 ( q ; q ) n − k ( q ; q ) n + k . (3.15) Pro of. Clearly , ( 3.14 ) is the direct consequence of the Bailey lemma sp ecialized by the Bailey pair ( 3.9 ). Note that ( 3.15 ) comes from n X k =0 q k 2 ( q ; q ) 2 k ( q ; q ) n − k = 1 ( q ; q ) 2 n . F unding This researc h was supp orted by the Natural Science F oundation of Zhejiang Pro vince, Grant No. L Y24A010012 and the National Natural Science F oundation of China, Gran t No. 12471315. Declarations Conflict of in terest The author declares that they hav e no competing in terests related to this work. References [1] G.E. Andrews, On the q -analog of Kummer’s theorem and applications, Duk e Math J. 40 (1973) 525–528. [2] G.E. Andrews, q -Series: Their Dev elopmen t and Application in Analysis, Number Theory , Com binatorics, Physics, and Computer Algebra, CBMS Regional Conf. Series in Math., No. 66., Amer. Math. So c., Providence, RI, 1986. [3] G.E. Andrews, S.O. W arnaar, The pro duct of partial theta functions. A dv. in Appl. Math. 39(1) (2007) 116–120 . [4] G. Gasp er, M. Rahman, Basic Hyp ergeometric Series, 2nd edition, Cambridge Univ er- sit y Press, 2004. 14 [5] K. Ragha vend ar, Certain q -series identities and applications to Lecture hall t ype par- titions, J. Anal. 28 (2020) 209–223. https://doi.org/10.1007/s41478- 017- 0061- 6 [6] J. W ang, X.R. Ma, On the Andrews-W arnaar identities for partial theta functions, A dv. in Applied Math. 97 (2018) 36–53. [7] S.O. W arnaar, Partial theta functions. I. Bey ond the lost noteb o ok, Pro c. London Math. So c. 87(3) (2003) 363–395. 15