(q n a; q) = ( a) n q (n+1 2 ) (q/a; q)n (a; q). For convenience, we employ the following notation for multiple q-shifted factorial:

Transcription

1 ARCHIVUM MATHEMATICUM (BRNO) Tomus 45 (2009) SEVERAL q-series IDENTITIES FROM THE EULER EXPANSIONS OF (a; q) AND (a;q) Zhizheng Zhang 2 and Jizhen Yang Abstract In this paper we first give several operator identities which extend the results of Chen and Liu then make use of them to two q-series identities obtained by the Euler expansions of (a; q) and Several (a;q) q-series identities are obtained involving a q-series identity in Ramanujan s Lost Notebook Introduction Throughout this paper assume that 0 < q < and adopt the customary notation in [6] for q-series Let () (a; q) ( aq k ) k0 For any integer n the q-shifted factorial (a; q) n is given by (2) (a; q) n (a; q) (a ; q) The q-binomial coefficient is defined by [ ] n (3) k (q; q) k k We will also use frequently the following equations: (4) or (q/a; q) n ( a) n q (n+ 2 ) (q n a; q) (a; q) (5) (q n a; q) ( a) n q (n+ 2 ) (q/a; q)n (a; q) For convenience we employ the following notation for multiple q-shifted factorial: where n is an integer or (a a 2 a m ; q) n (a ; q) n (a 2 ; q) n (a m ; q) n 2000 Mathematics Subject Classification: primary 05A30; secondary 33D5 33D60 Key words and phrases: exponential operator operator identity q-series identity Received March Editor O Došlý

2 48 Z ZHANG AND J YANG and The q-differential operator D q and the q-shifted operator are defined by D q f(a) (f(a) f(aq)) a ηf(a) f(aq) respectively These two operators have been introduced in [9 0 ] due to Rogers who applied them for proving Roger-Ramanujan identities Built on the D q and η the operator appeared in the work of Roman [2] and will be denoted by θ η D q Two operators introduced in the papers [4] and [5] by Chen and Liu are the exponential operators constructed from D q and θ: (bd q ) n T (bd q ) and E(bθ) (bθ) n q (n 2) Then the following operator identities were obtained Theorem (Chen and Liu [4] and [5]) Let T (bd q ) and E(bθ) be respectively defined as above Then (6) T (bd q ) (at; q) (at bt; q) (7) (8) (9) T (bd q ) (as at; q) (abst; q) (as at bs bt; q) E(bθ) (at; q) (at bt; q) E(bθ) (as at; q) (as at bs bt; q) (abst/q; q) By these operator identities in [ ] a lot of q-series identities can be derived In [4] Theorem was extended Applying generalized operator identities to some terminating basic hypergeometric series and q-integrals we obtain some summation formulas involving 3 φ 2 summation and some q-integrals involving 3 φ 2 summation which extend some famous q-integrals such as the Ismail-Stanton-Viennot integral Gasper integral formula In this paper we first give several operator identities which extend the results of Chen and Liu Then making use of them to two q-series identities obtained by the Euler expansions of (a; q) and (a;q) several q-series identities are obtained involving a q-series identity in Ramanujan s Lost Notebook Throughout this paper n and k are two nonnegative integers

3 SEVERAL q-series IDENTITIES 49 2 Several operator identities First of all by induction we give a result which is to be used later Lemma 2 We have (0) () θ n a k D n q a k 0 if n > k ( ) n (q k ; q) n a k n if n k 0 if n > k (q k n+ ; q) n a k n if n k (2) (3) θ n a k ( q) n (q k ; q) n a (k+n) D n q a k q (n 2) kn (q k ; q) n a (k+n) (4) In [3] we obtained the following operator identity E(dθ) a k (at as; q) a k (as at ds dt; q) (adst/q) 3φ 2 This identity can be rewritten by ( ) q k q/as q/at 0 q 2 /adst ; q q (5) E(dθ) a k (at as; q) a k (at as dt ds; q) (adst/q; q) [ ] k (q/at q/as; ( q)j j (q 2 q (j+ 2 ) kj /adst; q Next we give three operator identities which are extensions of Theorem They specialize to Theorem by setting k 0 Theorem 22 We have (6) (7) a k (at as; q) E(dθ) a k (at as; q) a k (at as dtqk dsq k ; q) (adstq k ; q) a k (at as dt ds; q) (adst/q; q) a k (adst; q) (at as dt ds; q) (q k ; q) n [ ] k (at as; q)j ( d j (adst; q a (q k adstq k ( ; q) n (q dtq k dsq k q (n+ 2 ) d ) n ; q) n a (adst/q; q) n+k q (n+ 2 ) ( d ) n (dt ds; q) n+k a

4 50 Z ZHANG AND J YANG where d/a < and a k (at as; q) a k (adtsq k ; q) (at as dtq k dsq k ( ) n (q k q +k /adst; q) n ; q) (q q +k /dt q +k /ds; q) n ( ) kq ( k+ 2 ) (adts; q) (q k ; q) n (q/adst; q) n+k (8) ( q) n d (at as dt ds; q) (q/dt q/ds; q) n+k Proof We note that the rule of Leibniz for D n q (see [4]) (9) D n q f(a)g(a) n [ ] n q k(k n) Dq k f(a)d k g(aq k ) k k0 Therefore we have a k (at as; q) d n d n nj d n Dq n ] n [ n q j(j n n [ ] n (q; q j(j n) q)k j d n q j(j n) d j (q; q) k (q; q (q; q) k j a k j [ ] k (d a k j a a k [ ] k (d j a a k (at as; q) D j qa k D n j q a k j Dq n j (q; q) k j (atq j asq j ; q) [ ] n (q; q)k a k j Dq n j j (q; q) k j (dq j D q ) n T (dq j D q (atq j asq j ; q) (atq j asq j ; q) d n q nj D n q (atq j asq j ; q) (atq j asq j ; q) (atq j asq j ; q) [ ] k (d (adstq a k j ; q) j a (atq j asq j dt ds; q) a k (adst; q) (at as dt ds; q) [ ] k (at as; q)j ( d j (adst; q a We obtain (6) (7) and (8) can be proved similarly The proof is completed Taking s 0 in Theorem 22 we have

5 SEVERAL q-series IDENTITIES 5 Corollary 23 (20) (2) a k a k (at; q) (at dt; q) [ ] k ( d j (at; q j a) E(dθ) a k (at; q) a k (at dtq k ; q) (q k ; q) n q (n+ 2 ) ( d (dt; q) n+k a ) n where d/a < and (22) a k ( (at; q) adt ) kq ( k+ 2 ) (at dt; q) (q k ; q) n (q/dt; q) n+k ( q at ) n where q/at < Further setting k or t 0 in (2) we get the following results of Chen and Liu Corollary 24 (Chen and Liu [5]) (23) (24) E(bθ) a a E(bθ) a ( a; q) a ( a; q) ( ) n q (n+ 2 ) b n a n m0 ( b/a) m q (m+ 2 ) ( bq m+ ; q) 3 Several q-series identities from expansion of (a; q) From the Euler expansion of (a; q) (see [3]): ( ) n q (n 2) a n (a; q) the following identity can be verified (25) ( a+ ; q) a + a ( a; q)

6 52 Z ZHANG AND J YANG Theorem 3 We have (26) k+ ( adt; q) n ( aq dq; q) n (q (k+) q n /a q/at; q (q q n q j /adt; q a ( adt; q) ( aq dq; q) ( + a)( + d) + a + adt/q (q k q/at; q (dt (q; q (q k q/a q/at; q (q q 2 /adt; q q j Proof Multiply both sides in equation (25) by a k+ (at; q) Then (27) a k+ ( a+ at; q) a k (at; q) + a k ( a at; q) Apply the operator E(dθ) to its both sides with respect to the variable a Then E(dθ) a k+ ( a+ at; q) E(dθ) a k (at; q) (28) + E(dθ) a k ( a at; q) Applying Theorem 22 and Corollary 23 we have E(dθ) a k+ ( a+ at; q) a k+ ( aqn+ at d+ dt; q) ( adt ; q) k+ (q (k+) q n /a q/at; q (q q n /adt; q q j E(dθ) a k (at; q) a k (at dt; q) E(dθ) a k ( a at; q) a k ( a at d dt; q) ( adt/q; q) (q k q/at; q (q; q (dt (q k q/a q/at; q (q q 2 /adt; q q j Substituting these three identities into (27) and then using (4) we obtain (3) after simplifying This completes the proof Taking k 0 in Theorem 3 we have Corollary 32 ( adt; q) n a ( adt; q) n + d( at/q) + ( aq dq; q) n ( aq; q) n ( dq; q) n+ (29) Taking t 0 in Corollary 32 we have ( adt; q) ( aq dq; q) + + a

7 SEVERAL q-series IDENTITIES 53 Corollary 33 a + d ( aq dq; q) n (30) Taking k in Theorem 3 we have ( aq; q) n ( dq; q) n+ + ( aq dq; q) + + a Corollary 34 a 2 ( adt; q) n q ( adt; q) n ad( + q)(at q) ( aq dq; q) n ( aq; q) n ( dq; q) n+ (3) + d 2 (at q)(at q 2 ) ( + a)(aq + dq + adq2 adt) + dq Taking t 0 in Corollary 34 we have Corollary 35 (32) a 2 ( aq dq; q) n + ad( + q) + d 2 q 2 ( aq; q) n ( dq; q) n+2 ( + a)(a + d + adq) + dq ( adt; q) n ( aq; q) n ( dq; q) n+2 ( adt; q) (aq + dq adt) ( aq dq; q) (a + d) ( aq dq; q) i0 ( aq; q) n ( dq; q) n+ Theorem 36 We have ( dtq k ( adtq k ; q) n ) ( aq dq k+ (q k adt+k ; q) i ; q) n (q d+k+ dtq k q (i+ 2 ) ( d ) i ; q) i a (33) where d/a < a ( adtq k ; q) ( aq dq k+ ; q) + ( + a ) i0 Proof (25) can be rewritten by (34) i0 (q k+ ; q) i (q dtq k+ q (i+ 2 ) ( d ) i ; q) i a (q k+ adtq k ; q) i (q dq k+ dtq k+ ; q) i q (i+ 2 ) ( d a) i a k ( a+ at; q) a (k+) (at; q) + a (k+) ( a at; q)

8 54 Z ZHANG AND J YANG Apply the operator E(dθ) to its both sides with respect to the variable a Then E(dθ) a k ( a+ at; q) E(dθ) a (k+) (at; q) + E(dθ) a (k+) ( a at; q) By Theorem 22 and Corollary 23 we have E(dθ) a k ( a+ at; q) a k ( aqn+ at d+k+ dtq k ; q) ( adt+k ; q) (q k adt+k ( ; q) i (q d+k+ dtq k q (i+ 2 ) d i ; q) i a) i0 E(dθ) a (k+) (at; q) a (k+) (at dtq k+ ; q) (q k+ ( ; q) i (q dtq k+ q (i+ 2 ) d i ; q) i a) i0 E(dθ) a (k+) ( a at; q) a (k+) ( a at dqk+ dtq k+ ; q) ( adtq k ; q) (q k+ adtq k ; q) i (q dq k+ dtq k+ q (i+ 2 ) ( d i ; q) i a) i0 Substituting these three identities into (34) and then using (4) we obtain the theorem Taking k 0 in Theorem 36 we have Corollary 37 ( adt; q) n ( dt) a ( adt; q) ( aq dq; q) n ( aq dq; q) (35) + ( + a ) where d/a < i0 i0 q (i+ 2 ) ( d ) i (dtq; q) i a ( adt; q) i ( dq dtq; q) i q (i+ 2 ) ( d a) i Taking t 0 in Corollary 37 we obtain the following Ramanujan s identity Corollary 38 a ( aq dq; q) n ( aq dq; q) (36) where d/a < + ( + a ) ( ) n q (n+ 2 ) (d/a) n ( ) n q (n+ 2 ) (d/a) n ( dq; q) n

9 SEVERAL q-series IDENTITIES 55 Note This identity is a formula in Ramanujan s Lost Notebook and its proofs are given by Andrews [ 2] In [5] Chen and Liu gave also a simple proof by using the method of the operator identity 4 Several q-series identities from expansion of (a;q) From the Euler expansion of (a;q) (see [3]): a n (a; q) the following identity can be verified (37) (a a a ; q) (a; q) From this identity by the operator T (dd q we can obtain the following results Theorem 4 We have k+ [ ] k + (a; q) ( n+j d (d; q) n (at; q a j (adt; q) n+j a (38) a (a d; q) (adt; q) Proof By (37) we have (39) a k+ (a at; q) [ ] ( j k d (at; q j a) a k (a at; q) ak (at; q) [ ] k (a at; q)j ( d j (adt; q a Apply the operator to its both sides with respect to the variable a Then a k+ a k (a at; q) (a at; q) (40) a k (at; q) By Theorem 22 and Corollary 23 we have a k+ (a a k+ (adt ; q) at; q) (a at d dt; q) k+ [ ] k + (a at; q) ( j d (4) j (adt ; q a (42) a k (a at; q) a k (adt; q) (a at d dt; q) [ ] k (a at; q)j ( d j (adt; q a

10 56 Z ZHANG AND J YANG and (43) a k a k (at; q) (at dt; q) [ ] k ( d j (at; q j a) Substituting these three identities into (40) and then using (4) we obtain the theorem Taking k 0 in Theorem 4 we have Corollary 42 (a d; q) n (44) a (a; q) n+ + d( at) (d; q) n (a d; q) (adt; q) n (adt; q) n+ (adt; q) Specially taking t 0 we have Corollary 43 (45) a (a d; q) n + d (d; q) n (a; q) n+ (a d; q) Taking k in Theorem 4 we have Corollary 44 (46) a 2 (a d; q) n (adt; q) n + ad( + q)( at) Specially taking t 0 we have Corollary 45 a 2 (47) (a d; q) n + ad( + q) Theorem 46 We have (a d; q) n (adt; q) n (48) + d 2 ( at)( atq) (a; q) n+ (d; q) n (adt; q) n+ (a; q) n+2 (d; q) n (adt; q) n+2 ( a)( at) a + d (a d; q) (a + d adt) adt (adt; q) (a; q) n+ (d; q) n + d 2 (a + d)( (a d; q) ) ad qk+ d (q k ; q (q; q + (a d; q) d (adt; q) (q n /adt; q) k+j (q/dt q n /d; q) k+j ( q (q k+ ; q (q; q (q/adt; q) k++j (q/d q/dt; q) k++j ( q (q k+ ; q (q; q (a; q) n+2 (d; q) n ( (q/dt; q) k++j (q/at) k++j

11 SEVERAL q-series IDENTITIES 57 where q/at < Proof We rewrite (37) into the following form: a k (49) (a a (k+) a (k+) at; q) (a at; q) (at; q) Apply the operator to its both sides with respect to the variable a Then a k a (k+) (a at; q) (a at; q) (50) a (k+) (at; q) By Theorem 22 and Corollary 23 we have a k (a ( d) k q (k+ 2 ) (adt ; q) at; q) (a at d dt; q) (q k ; q (q n /adt; q) k+j (5) (q; q (q/dt q n ( q /d; q) k+j (52) and (53) a (k+) (a at; q) (q k+ ; q (q; q a (k+) (at; q) ( d) (k+) q (k+2 2 ) (adt; q) (a at d dt; q) (q/adt; q) k+j+ (q/d q/dt; q) k+j+ ( q ( ) (k+)q ( k+2 2 ) adt (at dt; q) (q k+ ; q (q; q (q/dt; q) k+j+ ( q at Substituting these three identities into (50) and then using (4) we obtain the theorem Taking k 0 in Theorem 46 we have Corollary 47 (a d; q) n q (q/adt; q+ ( q (adt; q) n d (q/d q/dt; q+ (54) where q/at < + (a d; q) d (adt; q) ( ( q + (q/dt; q+ at

12 58 Z ZHANG AND J YANG Acknowledgement This research is supported by the National Natural Science Foundation of China (Grant No ) References [] Andrews G E Ramanujan s Lost Notebook I Partial θ-function Adv Math 4 (98) [2] Andrews G E Ramanujan: Essays and Surveys ch An introduction to Ramanujan s Lost Notebook pp [3] Andrews G E Askey R Roy R Special Function Cambridge University Press Cambridge 999 [4] Chen W Y C Liu Z-G Parameter augmentation for basic hypergeometric series II J Combin Theory Ser A 80 (997) [5] Chen W Y C Liu Z-G Mathematical essays in honor of Gian-Carlo Rota ch Parameter augmentation for basic hypergeometric series I pp 29 Birkhäuser Basel 998 [6] Gasper G G Rahman M Basic Hypergeometric Series Encyclopedia of Mathematics and Its Applications Second edition vol 96 Cambridge University Press 2004 [7] Liu Z-G A new proof of the Nassrallah-Rahman integral Acta Math Sinica 4 (2) (998) [8] Liu Z-G Some operator identities and q-series transformation formulas Discrete Math 265 (2003) 9 39 [9] Rogers L J On the expansion of some infinite products Proc London Math Soc 24 (893) [0] Rogers L J Second Memoir on the expansion of certain infinite products Proc London Math Soc 25 (894) [] Rogers L J Third Memoir on the expansion of certain infinite products Proc London Math Soc 26 (896) 5 32 [2] Roman S More on the umbral calculus with emphasis on the q-umbral calculus J Math Anal Appl 07 (985) [3] Zhang Z Z Liu M X Applications of operator identities to the multiple q-binomial theorem and q-gauss summation theorem Discrete Math 306 (2006) [4] Zhang Z Z Wang J Two operator identities and their applications to terminating basic hypergeometric series and q-integrals J Math Anal Appl 32 (2) (2005) Center of Combinatorics and LPMC Nankai University Tianjin P R China 2 Department of Mathematics Luoyang Normal University Luoyang P R China zhzhzhang-yang@63com