q-Analogues of π-Series by Applying Carlitz Inversions to q-Pfaff-Saalsch{ü}tz Theorem
aa r X i v : . [ m a t h . N T ] F e b q -ANALOGUES OF π -SERIES BY APPLYING CARLITZINVERSIONS TO q -PFAFF-SAALSCH ¨UTZ THEOREM XIAOJING CHEN AND WENCHANG CHU
Abstract.
By applying multiplicate forms of the Carlitz inverse series rela-tions to the q -Pfaff-Saalsch¨utz summation theorem, we establish twenty fivenonterminating q -series identities with several of them serving as q -analoguesof infinite series expressions for π and 1 /π , including some typical ones discov-ered by Ramanujan (1914) and Guillera. Introduction and Motivation
Let N and N be the sets of natural numbers and nonnegative integers, respectively.For an indeterminate x , the Pochhammer symbol is defined by( x ) ≡ x ) n = x ( x + 1) · · · ( x + n −
1) for n ∈ N with the following shortened multiparameter notation (cid:20) α, β, · · · , γA, B, · · · , C (cid:21) n = ( α ) n ( β ) n · · · ( γ ) n ( A ) n ( B ) n · · · ( C ) n . Analogously, the rising and falling q -factorials with the base q are given by ( x ; q ) = h x ; q i ≡ x ; q ) n = (1 − x )(1 − qx ) · · · (1 − q n − x ) h x ; q i n = (1 − x )(1 − q − x ) · · · (1 − q − n x ) ) for n ∈ N . Then the Gaussian binomial coefficient can be expressed as (cid:20) mn (cid:21) = ( q ; q ) m ( q ; q ) n ( q ; q ) m − n = ( q m − n +1 ; q ) n ( q ; q ) n where m, n ∈ N . When | q | <
1, the infinite product ( x ; q ) ∞ is well defined. We have hence the q -gamma function [10, § q ( x ) = (1 − q ) − x ( q ; q ) ∞ ( q x ; q ) ∞ and lim q → − Γ q ( x ) = Γ( x ) . For the sake of brevity, the product and quotient of the q -shifted factorials will beabbreviated respectively to[ α, β, · · · , γ ; q ] n = ( α ; q ) n ( β ; q ) n · · · ( γ ; q ) n , (cid:20) α, β, · · · , γA, B, · · · , C (cid:12)(cid:12)(cid:12) q (cid:21) n = ( α ; q ) n ( β ; q ) n · · · ( γ ; q ) n ( A ; q ) n ( B ; q ) n · · · ( C ; q ) n . According to Bailey [1] and Gasper–Rahman [10], the q -series is defined by ℓ φ ℓ (cid:20) a , a , · · · , a ℓ b , · · · , b ℓ (cid:12)(cid:12)(cid:12) q ; z (cid:21) = ∞ X n =0 (cid:20) a , a , · · · , a ℓ q, b , · · · , b ℓ (cid:12)(cid:12)(cid:12) q (cid:21) n z n . Mathematics Subject Classification.
Primary 33D15, Secondary 05A30, 11B65, 33D05.
Key words and phrases.
Basic hypergeometric series; The q -Pfaff-Saalsch¨utz summation theorem;Carlitz inverse series relations; Bisection series; Ramanujan–like series for π and 1 /π .Correspondence: [email protected] and [email protected]. Xiaojing Chen and Wenchang Chu
This series is well defined when none of the denominator parameters has the form q − m with m ∈ N . If one of the numerator parameters has the form q − m with m ∈ N , the series is terminating (in that case, it is a polynomial of z ). Otherwise,the series is said nonterminating, where we assume that 0 < | q | < q -analogues of the Gould–Hsu [11] inversions, Carlitz [3] found, in 1973, awell–known pair of inverse series relations, which can be reproduced as follows. Let { a k , b k } k ≥ be two sequences such that the ϕ -polynomials defined by ϕ ( x ; 0) ≡ ϕ ( x ; n ) = n − Y k =0 ( a k + xb k ) for n = 1 , , · · · differ from zero at x = q − m for m ∈ N . Then the following inverse relations hold f ( n ) = n X k =0 ( − k (cid:20) nk (cid:21) ϕ ( q − k ; n ) g ( k ) , (1) g ( n ) = n X k =0 ( − k (cid:20) nk (cid:21) q ( n − k ) a k + q − k b k ϕ ( q − n ; k + 1) f ( k ) . (2)Alternatively, if the ϕ -polynomials differ from zero at x = q m for m ∈ N , Carlitzdeduced, under the base change q → q − , another equivalent pair f ( n ) = n X k =0 ( − k (cid:20) nk (cid:21) q ( n − k ) ϕ ( q k ; n ) g ( k ) , (3) g ( n ) = n X k =0 ( − k (cid:20) nk (cid:21) a k + q k b k ϕ ( q n ; k + 1) f ( k ) . (4)These inverse series relations have been shown by and Chu [5, 6] to be very usefulin proving terminating q -series identities. Among numerous q -series identities, the q -Pfaff–Saalsch¨utz theorem (cf. [10, II-12] for the terminating balanced series isfundamental: φ (cid:20) q − n , a, bc, q − n ab/c (cid:12)(cid:12)(cid:12) q ; q (cid:21) = (cid:20) c/a, c/bc, c/ab (cid:12)(cid:12)(cid:12) q (cid:21) n . (5)As a warm–up, we illustrate how to derive the q -Dougall sum by making use ofCarlitz’ inversions. Observe that (5) is equivalent to φ (cid:20) q − n , q n a, qa/bdqa/b, qa/d (cid:12)(cid:12)(cid:12) q ; q (cid:21) = (cid:16) qabd (cid:17) n (cid:20) b, dqa/b, qa/d (cid:12)(cid:12)(cid:12) q (cid:21) n which can be rewritten as a q -binomial sum n X k =0 ( − k (cid:20) nk (cid:21) q ( n − k )( q k a ; q ) n (cid:20) a, qa/bdqa/b, qa/d (cid:12)(cid:12)(cid:12) q (cid:21) k = (cid:16) qabd (cid:17) n (cid:20) a, b, dqa/b, qa/d (cid:12)(cid:12)(cid:12) q (cid:21) n q ( n ) . This matches exactly (3) under the specifications f ( n ) = (cid:16) qabd (cid:17) n (cid:20) a, b, dqa/b, qa/d (cid:12)(cid:12)(cid:12) q (cid:21) n q ( n ) ,g ( k ) = (cid:20) a, qa/bdqa/b, qa/d (cid:12)(cid:12)(cid:12) q (cid:21) k and ϕ ( x ; n ) = ( ax ; q ) n . Then the dual relation corresponding to (4) reads as n X k =0 ( − k (cid:20) nk (cid:21) − q k a ( q n a ; q ) k +1 (cid:16) qabd (cid:17) k (cid:20) a, b, dqa/b, qa/d (cid:12)(cid:12)(cid:12) q (cid:21) k q ( k ) = (cid:20) a, qa/bdqa/b, qa/d (cid:12)(cid:12)(cid:12) q (cid:21) n . -Analogues of π -Series 3 This is equivalent to the q -Dougall sum (cf. [10, II-21]): φ (cid:20) a, q √ a, − q √ a, b, d, q − n √ a, −√ a, qa/b, qa/d, q n +1 a (cid:12)(cid:12)(cid:12) q ; q n +1 abd (cid:21) = (cid:20) qa, qa/bdqa/b, qa/d (cid:12)(cid:12)(cid:12) q (cid:21) n . For a = b = d = q / , the limiting case n → ∞ of the last formula becomes1Γ q ( ) = ∞ X k =0 ( − k ( q / ; q ) k ( q ; q ) k − q k + − q q k which reduces, for q → − , to the following infinite series expression for π π = ∞ X k =0 ( − k ( ) k (1) k (cid:8) k (cid:9) as recorded in one of Ramanujan’s letters to Hardy [19]. More difficult formulaefor 1 /π were subsequently discovered by Ramanujan [20, 1914], where 17 similarseries representations were announced. Three of them are reproduced as follows:4 π = ∞ X k =0 " , , , , k k k . π = ∞ X k =0 " , , , , k k ( − k . π = ∞ X k =0 " , , , , k k k . For their proofs and recent developments, the reader can consult the papers byBaruah-Berndt-Chan [2], Guillera [12–14] and Chu et al [7, 9].Recently, there has been a growing interest in finding q -analogues of Ramanujan–like series (cf. [4, 8, 15–18]). Following the procedure just described, the aim of thispaper is to show systematically q -analogues of π -related series by applying the mul-tiplicate form of Carlitz inverse series relations to the q -Pfaff–Saalsch¨utz summationtheorem. In the next section, we shall derive, by employing the duplicate inver-sions, twenty q -series identities including q -analogues of the afore displayed threeseries of Ramanujan. Then in section 3, the triplicate inversions will be utilizedto establish five q -series identities. By applying the bisection series method to tworesulting series, q -analogues are established also for the following two remarkableseries discovered by Guillera [12, 13]:2 √ π = ∞ X k =0 (cid:16) − (cid:17) k " , , , , k (cid:8) k (cid:9) . √ π = ∞ X k =0 (cid:16) − (cid:17) k " , , , , k (cid:8)
15 + 154 k (cid:9) . Duplicate Inverse Series Relations
Denote by ⌊ x ⌋ the integer part for a real number x . Then for all the n ∈ N , thereholds the equality n = (cid:4) n (cid:5) + (cid:4) n (cid:5) . Xiaojing Chen and Wenchang Chu
According to this partition, we shall reformulate (5) in three different manners.Their dual relations will lead us to q -series counterparts for several remarkableinfinite series expressions of π and 1 /π .2.1. According to the q -Pfaff–Saalsch¨utz formula (5), it is not hard to verify that φ (cid:20) q − n , a, cq −⌊ n ⌋ ae, q −⌊ n +12 ⌋ c/e (cid:12)(cid:12)(cid:12) q ; q (cid:21) = (cid:20) q −⌊ n ⌋ e, q −⌊ n ⌋ ae/cq −⌊ n ⌋ ae, q −⌊ n ⌋ e/c (cid:12)(cid:12)(cid:12) q (cid:21) n which is equivalent to the binomial sum n X k =0 ( − k (cid:20) nk (cid:21) ( q − k /ae ; q ) ⌊ n ⌋ ( q − k e/c ; q ) ⌊ n +12 ⌋ (cid:20) a, cae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) k q ( k +12 )= (cid:20) e, ae/cae (cid:12)(cid:12)(cid:12) q (cid:21) ⌊ n +12 ⌋ (cid:20) q/e, qc/aeqc/e (cid:12)(cid:12)(cid:12) q (cid:21) ⌊ n ⌋ . Observing that this equation matches exactly to (1) specified by f ( k ) = (cid:20) e, ae/cae (cid:12)(cid:12)(cid:12) q (cid:21) ⌊ k +12 ⌋ (cid:20) q/e, qc/aeqc/e (cid:12)(cid:12)(cid:12) q (cid:21) ⌊ k ⌋ ,g ( k ) = (cid:20) a, cae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) k q ( k +12 ) ,ϕ ( x ; n ) = ( qx/ae ; q ) ⌊ n ⌋ ( ex/c ; q ) ⌊ n +12 ⌋ ;we may state the dual relation corresponding to (2) as the proposition. Proposition 1 (Terminating reciprocal relation) . (cid:20) a, cae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) n = X k ≥ (cid:20) n k (cid:21) (1 − q − k e/c ) q (1+2 k )( k − n ) ( q − n /ae ; q ) k ( q − n e/c ; q ) k +1 (cid:20) e, ae/cae (cid:12)(cid:12)(cid:12) q (cid:21) k (cid:20) q/e, qc/aeqc/e (cid:12)(cid:12)(cid:12) q (cid:21) k − X k ≥ (cid:20) n k + 1 (cid:21) (1 − q − k /ae ) q (1+ k )(1+2 k − n ) ( q − n /ae ; q ) k +1 ( q − n e/c ; q ) k +1 (cid:20) e, ae/cae (cid:12)(cid:12)(cid:12) q (cid:21) k +1 (cid:20) q/e, qc/aeqc/e (cid:12)(cid:12)(cid:12) q (cid:21) k . The two sums just displayed are, in fact, balanced φ -series, which do not admitclosed forms. However their combination does have a closed form. That is thereason why we call the last relation reciprocal.Letting n → ∞ in Proposition 1 and then applying the Weierstrass M -test onuniformly convergent series (cf. Stromberg [21, § (cid:20) a, cae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) ∞ = X k ≥ − q k c/e ( q ; q ) k (cid:20) e, ae/cae (cid:12)(cid:12)(cid:12) q (cid:21) k (cid:20) q/e, qc/aeqc/e (cid:12)(cid:12)(cid:12) q (cid:21) k q k − k ( ac ) k + ce X k ≥ − ae/q ( q ; q ) k +1 (cid:20) e, ae/cae/q (cid:12)(cid:12)(cid:12) q (cid:21) k +1 (cid:20) q/e, qc/aeqc/e (cid:12)(cid:12)(cid:12) q (cid:21) k q k ( ac ) k . By unifying the two sums together, we find the following theorem.
Theorem 2 (Nonterminating series identity) . (cid:20) a, cae, c/e (cid:12)(cid:12)(cid:12) q (cid:21) ∞ = ∞ X k =0 ( ac ) k ( q ; q ) k (cid:20) e, ae/cae (cid:12)(cid:12)(cid:12) q (cid:21) k (cid:20) q/e, qc/aec/e (cid:12)(cid:12)(cid:12) q (cid:21) k q k − k × (cid:26) q k c (1 − q k e )(1 − q k ae/c ) e (1 − q k )(1 − q k c/e ) (cid:27) . -Analogues of π -Series 5 We highlight two important corollaries about product of reciprocal q -gamma func-tions. Their limiting case q → − will yield infinite series for π and 1 /π . Corollary 3 ( a = q λ and c = e = q − λ in Theorem 2) . q (1 + λ )Γ q (2 − λ ) = ∞ X k =0 (cid:2) q λ , q λ , q − λ , q − λ ; q (cid:3) k ( q ; q ) k ( q ; q ) k q k + k × (cid:26) − (1 − q − k )(1 − q k )(1 − q λ + k )(1 − q − λ + k ) (cid:27) . Corollary 4 ( a = c = q and e = q λ in Theorem 2) . Γ q ( λ )Γ q (1 − λ ) = ∞ X k =0 q k + k ( q λ ; q ) k ( q − λ ; q ) k ( q ; q ) k (cid:26) − q k − q λ + k − − q λ + k − q λ − − k (cid:27) . By properly choosing special values of a, c and e , we find ten interesting q -seriesidentities, that correspond to the classical series with convergence rate “ ”. A1 . For the series discovered by Ramanujan [20]4 π = ∞ X k =0 (cid:20) , , , , (cid:21) k k k , we recover, by letting λ = 1 / q -analogue (cf. Chen–Chu [4, Example 38] and Guo [16, Equation 1.6])1Γ q ( ) = ∞ X k =0 q k ( q / ; q ) k ( q ; q ) k ( q ; q ) k q k +1 / − q k +1 / (1 − q )(1 + q k +1 / ) . A different, but simpler q -analogue can be found in Guo–Liu [17, Equation 3] andChen–Chu [4, Example 4]: ∞ X k =0 − q k +1 − q ( q ; q ) k ( q ; q ) k ( q ; q ) k q k = 1Γ q ( ) . A2 . For λ = 1 /
3, we get, from Corollary 3, the following series1Γ q ( )Γ q ( ) = ∞ X k =0 q k + k (cid:2) q / , q / , q / , q / ; q (cid:3) k ( q ; q ) k ( q ; q ) k (cid:26) − (1 − q − k )(1 − q k +1 )(1 − q k + )(1 − q k + ) (cid:27) which gives a q -analogue of the series9 √ π = ∞ X k =0 " , , , , , , k k + 27 k k . A3 . For λ = 1 /
4, we have, from Corollary 3, the following series due to Guo andZudilin [18, Equation 1.6]1Γ q ( )Γ q ( ) = ∞ X k =0 q k + k h q , q , q , q ; q i k ( q ; q ) k ( q ; q ) k (cid:26) − (1 − q − k )(1 − q k +1 )(1 − q k + )(1 − q k + ) (cid:27) which offers a q -analogue of the series8 √ π = ∞ X k =0 " , , , , , , k k + 48 k k . Xiaojing Chen and Wenchang Chu A4 . For λ = 1 /
6, we find, from Corollary 3, the following series1Γ q ( )Γ q ( ) = ∞ X k =0 q k + k (cid:2) q / , q / , q / , q / ; q (cid:3) k ( q ; q ) k ( q ; q ) k (cid:26) − (1 − q − k )(1 − q k +1 )(1 − q k + )(1 − q k + ) (cid:27) which provides a q -analogue of the series18 π = ∞ X k =0 " , , , , , , k k + 108 k k . A5 . Letting λ = 1 / q (cid:0) (cid:1) = ∞ X k =0 q k + k ( q / ; q ) k ( q ; q ) k (1 + 2 q k +1 / )which is a q -analogue of the series π ∞ X k =0 " , , k (cid:18) (cid:19) k . A6 . Letting λ = 1 / q (cid:18) (cid:19) Γ q (cid:18) (cid:19) = ∞ X k =0 q k + k ( q / ; q ) k ( q / ; q ) k ( q ; q ) k (cid:26) − q k − q k + − − q k + − q − k − (cid:27) which gives a q -analogue of the series4 π √ ∞ X k =0 " , , , , , , k k + 27 k k . A7 . Letting λ = 1 / q (cid:18) (cid:19) Γ q (cid:18) (cid:19) = ∞ X k =0 q k + k ( q / ; q ) k ( q / ; q ) k ( q ; q ) k (cid:26) − q k − q k + − − q k + − q − k − (cid:27) which results in a q -analogue of the series10 π = ∞ X k =0 " , , , , , , k
31 + 108 k + 108 k k . A8 . By specifying a = q, c = q / and e = q / in Theorem 2, we findΓ q ( )Γ q ( ) = ∞ X k =0 q k + k ( q / ; q ) k ( q / ; q ) k ( q / ; q ) k ( q ; q ) k q k + − q k +1 − q which corresponds to the classical series √ ( )2 π = ∞ X k =0 " , , , , k k k . A9 . By specifying a = c = q / and e = q / in Theorem 2, we haveΓ q ( )Γ q ( ) = ∞ X k =0 q k ( k − ) ( q / ; q ) k ( q / ; q ) k ( q / ; q ) k ( q ; q ) k q k + − q k + (1 − q )(1 + q ) -Analogues of π -Series 7 which corresponds to the classical series2Γ ( )3Γ ( ) = ∞ X k =0 " , , , , k k k ⇐⇒ ( )Γ ( ) = ∞ X k =0 " , , , , k k . A10 . By specifying a = c = q / and e = q / in Theorem 2, we findΓ q ( )Γ q ( ) = ∞ X k =0 q k ( k + ) ( q / ; q ) k ( q / ; q ) k ( q / ; q ) k ( q ; q ) k (1 + q )(1 + q k + − q k + )1 − q which corresponds to the classical seriesΓ ( )8Γ ( ) = ∞ X k =0 " , , , , k k k . φ (cid:20) q − n , q ⌊ n ⌋ a, cae, q −⌊ n +12 ⌋ c/e (cid:12)(cid:12)(cid:12) q ; q (cid:21) = (cid:20) q −⌊ n ⌋ e, ae/cq −⌊ n ⌋ e/c, ae (cid:12)(cid:12)(cid:12) q (cid:21) n . By making use of the factorial expression( q − k y ; q ) ⌊ n +12 ⌋ q ⌊ n +12 ⌋ k = h q k /y ; q i ⌊ n +12 ⌋ ( − y ) ⌊ n +12 ⌋ q ( ⌊ n +12 ⌋ ) , we can reformulate the last equality as the q -binomial identity: n X k =0 ( − k (cid:20) nk (cid:21) q ( n − k )( q k a ; q ) ⌊ n ⌋ h q k c/e ; q i ⌊ n +12 ⌋ (cid:20) a, cae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) k = ( − ⌊ n +12 ⌋ q ( n ) − ( ⌊ n +12 ⌋ ) c n ( e ; q ) ⌊ n +12 ⌋ e ⌊ n +12 ⌋ (cid:20) ae/cae (cid:12)(cid:12)(cid:12) q (cid:21) n (cid:20) q/e, aqc/e (cid:12)(cid:12)(cid:12) q (cid:21) ⌊ n ⌋ . Since the last equation matches exactly to (3) specified by f ( k ) = ( − ⌊ k +12 ⌋ q ( k ) − ( ⌊ k +12 ⌋ ) c k ( e ; q ) ⌊ k +12 ⌋ e ⌊ k +12 ⌋ (cid:20) ae/cae (cid:12)(cid:12)(cid:12) q (cid:21) k (cid:20) q/e, aqc/e (cid:12)(cid:12)(cid:12) q (cid:21) ⌊ k ⌋ ,g ( k ) = (cid:20) a, cae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) k and ϕ ( x ; n ) = ( ax ; q ) ⌊ n ⌋ h cx/e ; q i ⌊ n +12 ⌋ ;the dual relation corresponding to (4) is given in the proposition. Proposition 5 (Terminating reciprocal relation) . (cid:20) a, cae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) n = X k ≥ q k − k (cid:20) n k (cid:21) (1 − q k c/e )( − k c k ( q n a ; q ) k h q n c/e ; q i k +1 ( e ; q ) k e k (cid:20) ae/cae (cid:12)(cid:12)(cid:12) q (cid:21) k (cid:20) q/e, aqc/e (cid:12)(cid:12)(cid:12) q (cid:21) k + X k ≥ q k k (cid:20) n k + 1 (cid:21) (1 − aq k +1 )( − k c k +1 ( q n a ; q ) k +1 h q n c/e ; q i k +1 ( e ; q ) k +1 e k +1 (cid:20) ae/cae (cid:12)(cid:12)(cid:12) q (cid:21) k +1 (cid:20) q/e, aqc/e (cid:12)(cid:12)(cid:12) q (cid:21) k . Both sums just displayed can be expressed as terminating q -series, which do nothave closed forms. However their combination does have a closed form.Letting n → ∞ in Proposition 5 and then applying the Weierstrass M -test onuniformly convergent series, we get the limiting relation: (cid:20) a, cae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) ∞ = X k ≥ ( − k q k − k − q k c/e ( q ; q ) k c k ( e ; q ) k e k (cid:20) ae/cae (cid:12)(cid:12)(cid:12) q (cid:21) k (cid:20) q/e, aqc/e (cid:12)(cid:12)(cid:12) q (cid:21) k Xiaojing Chen and Wenchang Chu + ce X k ≥ ( − k q k k − aq k +1 ( q ; q ) k +1 c k ( e ; q ) k +1 e k (cid:20) ae/cae (cid:12)(cid:12)(cid:12) q (cid:21) k +1 (cid:20) q/e, aqc/e (cid:12)(cid:12)(cid:12) q (cid:21) k . By unifying the two sums together, we find the following theorem.
Theorem 6 (Nonterminating series identity) . (cid:20) a, cae, c/e (cid:12)(cid:12)(cid:12) q (cid:21) ∞ = ∞ X k =0 ( − c /e ) k ( q ; q ) k ( ae/c ; q ) k ( ae ; q ) k (cid:20) a, e, q/ec/e (cid:12)(cid:12)(cid:12) q (cid:21) k q k − k × (cid:26) q k c (1 − aq k +1 )(1 − q k e )(1 − q k ae/c ) e (1 − q k )(1 − q k c/e )(1 − aeq k ) (cid:27) . Two implications are given below about product of reciprocal q -gamma functions. Corollary 7 ( a = q λ and c = e = q − λ in Theorem 6) . q (1 + λ )Γ q (2 − λ ) = ∞ X k =0 ( − k ( q λ ; q ) k ( q ; q ) k (cid:20) q λ , q λ , q − λ q (cid:12)(cid:12)(cid:12) q (cid:21) k q k (3+3 k − λ ) × − q λ +3 k − q (cid:26) q − k (1 − q k )(1 − q k )(1 − q k )(1 − q − λ + k )(1 − q λ +2 k )(1 − q λ +3 k ) (cid:27) . Corollary 8 ( a = c = q and e = q λ in Theorem 6) . Γ q (1 + λ )Γ q (1 − λ ) = ∞ X k =0 ( − k (cid:2) q, q λ ; q (cid:3) k ( q ; q ) k ( q λ ; q ) k ( q λ ; q ) k q k (3+3 k − λ ) × (cid:26) q k − λ (1 − q k )(1 − q λ + k )(1 − q λ +2 k )(1 − q k )(1 − q − λ + k )(1 − q λ +2 k ) (cid:27) . Five q -series as well as their counterparts of classical series are exemplified as follows. B1 . For Ramanujan’s series [20]8 π = ∞ X k =0 (cid:16) − (cid:17) k (cid:20) , , , , (cid:21) k (cid:8) k (cid:9) , we recover, by letting λ = 1 / q -analogue (cf. Chenand Chu [4, Example 39])1Γ q ( ) = ∞ X k =0 ( − k ( q / ; q ) k ( q / ; q ) k ( q ; q ) k ( q ; q ) k q k / × (cid:26) (1 + q k +1 / ) (1 − q k +1 / ) − q k +1 / (1 − q k +1 / )(1 − q )(1 + q k +1 / ) (cid:27) . Guo and Zudilin [18, Equation 1.4] derived, by means of the WZ machinery, another q -analogue 1Γ q ( ) = ∞ X k =0 ( − k ( q / ; q ) k ( q / ; q / ) k ( q ; q ) k ( q ; q ) k q k / × (cid:26) − q k +1 / − q + q k +1 / (1 − q k +1 / )(1 − q )(1 + q k +1 / ) (cid:27) . This is another example (apart from A1 ) that there may exist different q -analoguesfor some classical series. -Analogues of π -Series 9 B2 . For λ = 1 /
2, we get, from Corollary 8, the seriesΓ q (cid:18) (cid:19) Γ q (cid:18) (cid:19) = ∞ X k =0 ( − k q k + k ( q ; q ) k ( q / ; q ) k ( q / ; q ) k ( q / ; q ) k ( q ; q ) k × (cid:26) q k + (1 − q k +2 )(1 − q k + )(1 − q k +1 )(1 − q k + ) (cid:27) which can also be obtained from Chu [8, Proposition 14: x = y = q ]. The aboveseries gives, in turn, a q -analogue of the classical series3 π ∞ X k =0 (cid:16) − (cid:17) k " , , , , k (cid:8) k + 20 k (cid:9) . B3 . For λ = 1 /
3, we have, from Corollary 8, the seriesΓ q (cid:18) (cid:19) Γ q (cid:18) (cid:19) = ∞ X k =0 ( − k +1 ( q ; q ) k +1 ( q / ; q ) k ( q / ; q ) k ( q ; q ) k ( q / ; q ) k +1 q k + k +1 × (cid:26) q k + )(1 − q − k − )(1 − q k +1 )(1 − q k +1 )(1 − q k + ) (cid:27) which offers a q -analogue of the classical series8 π √ ∞ X k =0 (cid:16) − (cid:17) k " , , , , k (cid:8) k + 30 k (cid:9) . B4 . For λ = 2 /
3, we obtain, from Corollary 8, the seriesΓ q (cid:18) (cid:19) Γ q (cid:18) (cid:19) = ∞ X k =0 ( − k ( q ; q ) k ( q / ; q ) k ( q / ; q ) k ( q ; q ) k ( q / ; q ) k q k + k × (cid:26) q k + (1 + q k + )(1 − q k + )(1 − q k +2 )(1 − q k +1 )(1 − q k + ) (cid:27) . which provides a q -analogue of the classical series20 π √ ∞ X k =0 (cid:16) − (cid:17) k " , , , , k (cid:8)
13 + 40 k + 30 k (cid:9) . B5 . In addition, by specifying a = q / , c = q / and e = q / in Theorem 6, wefind the following strange looking identityΓ q ( )Γ q ( )Γ q ( )Γ q ( ) = ∞ X k =0 ( − k h q , q ; q i k ( q ; q ) k ( q ; q ) k ( q ; q ) k q k × (cid:26) q k + (1 − q +2 k )(1 − q +3 k )(1 − q k )(1 − q +2 k ) (cid:27) which turns out to be a q -analogue of the classical series5 √ ∞ X k =0 (cid:16) − (cid:17) k " , , , , , , k (cid:8)
10 + 51 k + 60 k (cid:9) . φ (cid:20) q − n , q ⌊ n ⌋ a, q ⌊ n +12 ⌋ cae, qc/e (cid:12)(cid:12)(cid:12) q ; q (cid:21) = (cid:20) q −⌊ n ⌋ e, q −⌊ n +12 ⌋ ae/cq − n e/c, ae (cid:12)(cid:12)(cid:12) q (cid:21) n , which is equivalent to the binomial sum n X k =0 ( − k (cid:20) nk (cid:21) q ( n − k )( q k a ; q ) ⌊ n ⌋ ( q k c ; q ) ⌊ n +12 ⌋ (cid:20) a, cae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) k = q ⌊ n − n ⌋ a ⌊ n +12 ⌋ c ⌊ n ⌋ [ a, q/e, ae/c ; q ] ⌊ n ⌋ [ c, e, qc/ae ; q ] ⌊ n +12 ⌋ [ ae, qc/e ; q ] n . Now that this equation matches exactly to (3) specified by f ( k ) = q ⌊ k − k ⌋ a ⌊ k +12 ⌋ c ⌊ k ⌋ [ a, q/e, ae/c ; q ] ⌊ k ⌋ [ c, e, qc/ae ; q ] ⌊ k +12 ⌋ [ ae, qc/e ; q ] k ,g ( k ) = (cid:20) a, cae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) k and ϕ ( x ; n ) = ( ax ; q ) ⌊ n ⌋ ( cx ; q ) ⌊ n +12 ⌋ ;we have the dual relation corresponding to (4), which is evidenced below. Proposition 9 (Terminating reciprocal relation) . (cid:20) a, cae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) n = X k ≥ (cid:20) n k (cid:21) (1 − q k c ) q k − k ( ac ) k ( q n a ; q ) k ( q n c ; q ) k +1 [ a, q/e, ae/c ; q ] k [ c, e, qc/ae ; q ] k [ ae, qc/e ; q ] k − a X k ≥ (cid:20) n k + 1 (cid:21) (1 − q k +1 a ) q k +2 k ( ac ) k ( q n a ; q ) k +1 ( q n c ; q ) k +1 [ a, q/e, ae/c ; q ] k [ c, e, qc/ae ; q ] k +1 [ ae, qc/e ; q ] k +1 . Both sums just displayed are terminating q -series with nether of them admittingclosed forms. However their combination does have a closed form.Letting n → ∞ in Proposition 9 and then applying the Weierstrass M -test onuniformly convergent series, we get the limiting relation: (cid:20) a, cae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) ∞ = X k ≥ (1 − q k c ) q k − k ( ac ) k ( q ; q ) k [ a, q/e, ae/c ; q ] k [ c, e, qc/ae ; q ] k [ ae, qc/e ; q ] k − a X k ≥ (1 − q k +1 a ) q k +2 k ( ac ) k ( q ; q ) k +1 [ a, q/e, ae/c ; q ] k [ c, e, qc/ae ; q ] k +1 [ ae, qc/e ; q ] k +1 . By unifying the two sums together, we find the following theorem.
Theorem 10 (Nonterminating series identity) . (cid:20) a, qcae, qc/e (cid:12)(cid:12)(cid:12) q (cid:21) ∞ = ∞ X k =0 − q k c − c [ a, c, e, q/e, ae/c, qc/ae ; q ] k [ q, ae, qc/e ; q ] k q k − k ( ac ) k × (cid:26) − q k a (1 − q k +1 a )(1 − q k c )(1 − q k e )(1 − q k c/ae )(1 − q k c )(1 − q k )(1 − q k ae )(1 − q k c/e ) (cid:27) . Two special cases are recorded here about product of reciprocal q -gamma functions,which can be utilized to establish q -analogues of classical series for π and 1 /π . Corollary 11 ( a = q λ and c = e = q − λ in Theorem 10) . q ( λ )Γ q (1 − λ ) = ∞ X k =0 q k ( q λ ; q ) k ( q − λ ; q ) k ( q ; q ) k − q k +1 − λ − q -Analogues of π -Series 11 × (cid:26) − q k + λ (1 − q k +1+ λ )(1 − q k +1 − λ ) (1 − q k +1 − λ )(1 − q k +1 ) (cid:27) . Corollary 12 ( a = c = q and e = q λ in Theorem 10) . Γ q (1 + λ )Γ q (2 − λ ) = ∞ X k =0 − q k +1 − q (cid:2) q, q, q λ , q − λ , q λ , q − λ ; q (cid:3) k [ q, q λ , q − λ ; q ] k q k + k × (cid:26) − q k (1 − q k )(1 − q k )(1 − q λ + k )(1 − q − λ + k )(1 − q k )(1 − q k )(1 − q λ +2 k )(1 − q − λ +2 k ) (cid:27) . Five q -series as well as their counterparts of classical series are displayed as follows. C1 . Recall the following series of Ramanujan [20]:16 π = ∞ X k =0 (cid:20) , , , , (cid:21) k k k . By letting λ = 1 / q -analogue (cf. Chen and Chu [4,Example 40]) as follows1Γ q ( ) = ∞ X k =0 q k ( q / ; q ) k ( q ; q ) k − q k +1 / − q (cid:26) − q k +1 / (1 − q k +3 / )(1 + q k +1 / ) (1 − q k +1 / ) (cid:27) . C2 . For λ = 1 /
4, we get, from Corollary 11, the q -series identity1Γ q ( )Γ q ( ) = ∞ X k =0 − q k + − q ( q ; q ) k ( q ; q ) k ( q ; q ) k q k × (cid:26) − q k + (1 − q k + )(1 − q k + ) (1 − q k + )(1 − q k +1 ) (cid:27) . In order to simplify the last series, consider the series defined by ∞ X k =0 Λ( k ) , where Λ( k ) := ( − k − q k − q ( q ; q ) k ( q ; q ) k q k . Then its bisection series can be reformulated as ∞ X k =0 Λ( k ) = ∞ X k =0 (cid:8) Λ(2 k ) + Λ(2 k + 1) (cid:9) = ∞ X k =0 Λ(2 k ) n k + 1)Λ(2 k ) o = ∞ X k =0 − q k + − q ( q ; q ) k ( q ; q ) k q k (cid:26) − q k + (1 − q k + )(1 − q k + ) (1 − q k + )(1 − q k +1 ) (cid:27) . Now it is not hard to check that1 − q k + − q (cid:26) − q k + (1 − q k + )(1 − q k + ) (1 − q k + )(1 − q k +1 ) (cid:27) = 1 − q k + − q (cid:26) − q k + (1 − q k + )(1 − q k + ) (1 − q k + )(1 − q k +1 ) (cid:27) . We find the following simpler series (see Chen–Chu [4, Example 5] and Guo–Liu [17,Equation 4]) 1Γ q ( )Γ q ( ) = ∞ X k =0 ( − k − q k − q ( q ; q ) k ( q ; q ) k q k which results in a q -analogue of the classical one due to Guillera [13]2 √ π = ∞ X k =0 (cid:16) − (cid:17) k " , , , , k (cid:8) k (cid:9) . C3 . For λ = 1 /
2, we have, from Corollary 12, the q -series identityΓ q (cid:18) (cid:19) = ∞ X k =0 q k + k − q k +1 − q ( q ; q ) k ( q ; q ) k ( q ; q ) k ( q ; q ) k × (cid:26) − q k +1 (1 − q k + )(1 − q k +1 )(1 − q k +2 )(1 + q k + )(1 − q k + ) (1 − q k +1 ) (cid:27) which gives a q -analogue of the following series9 π ∞ X k =0 " , , , , , , k k + 75 k + 42 k k . We remark that the above q -series can also be derived by letting x = y = q inChu [8, Proposition 15]. C4 . Letting a = c = e = q / in Theorem 10, we derive the q -series identityΓ q ( )Γ q ( ) = ∞ X k =0 q k − k − q k + − q ( q ; q ) k ( q ; q ) k ( q ; q ) k ( q ; q ) k × (cid:26) − q k + (1 − q k + )(1 − q k + )(1 − q k + )(1 + q k + )(1 − q k +1 ) (1 − q k + ) (cid:27) which provides a q -analogue of the following series128 √ π Γ ( ) = ∞ X k =0 " , , , , , , k
17 + 396 k + 1392 k + 1344 k k . C5 . Letting a = c = e = q / in Theorem 10, we derive the q -series identityΓ q ( )Γ q ( ) = ∞ X k =0 q k + k − q k + − q ( q ; q ) k ( q ; q ) k ( q ; q ) k ( q ; q ) k × (cid:26) − q k + (1 − q k + )(1 − q k + )(1 − q k + )(1 + q k + )(1 − q k +1 ) (1 − q k + ) (cid:27) which serves as a q -analogue of the series64 √ π Γ ( ) = ∞ X k =0 " , , , , , , k (12 k + 5)(28 k + 15)64 k . Triplicate Inverse Series Relations
For all the n ∈ N , we have two equalities n = (cid:4) n (cid:5) + (cid:4) n (cid:5) = (cid:4) n (cid:5) + (cid:4) n (cid:5) + (cid:4) n (cid:5) . Then six dual relations can be established from (5). However, only two of themgive some interesting q -series identities. Five examples are illustrated in this sectionwithout reproducing the whole inversion procedure. -Analogues of π -Series 13 q -Pfaff–Saalsch¨utz theorem (5) φ (cid:20) q − n , a, cq −⌊ n ⌋ ae, q −⌊ n +13 ⌋ c/e (cid:12)(cid:12)(cid:12) q ; q (cid:21) = " q −⌊ n ⌋ e, q −⌊ n ⌋ ae/cq −⌊ n ⌋ ae, q −⌊ n ⌋ e/c (cid:12)(cid:12)(cid:12) q n we can derive three q -series identities corresponding to the classical series of con-vergence rate “ ”. D1 . For a = q / and c = e = q / , we have the corresponding identity1Γ q ( )Γ q ( ) = ∞ X k =0 q k + k − q h q , q ; q i k h q , q ; q i k +1 ( q ; q ) k ( q ; q ) k ( q ; q ) k +1 × (cid:26) − (1 − q − k )(1 − q k +1 )(1 − q k + )(1 − q k + ) + q k +1 (1 − q k + )(1 − q k + )(1 − q k +1 )(1 − q k +2 ) (cid:27) which gives a q -analogue of the classical series81 √ π = ∞ X k =0 (cid:16) (cid:17) k " , , , , , , k (cid:8)
20 + 243 k + 414 k (cid:9) . D2 . For a = c = q and e = q / , we get the corresponding identityΓ q (cid:18) (cid:19) Γ q (cid:18) (cid:19) = ∞ X k =0 q ( k +1)(2 k +2 / − q ( q ; q ) k ( q ; q ) k +1 ( q ; q ) k ( q ; q ) k ( q ; q ) k +1 × (cid:26) − (1 − q − k − )(1 − q k +1 )(1 − q k + ) + q k + (1 − q k + ) (1 − q k + )(1 − q k +2 ) (cid:27) which is a q -analogue of the following series8 π √ ∞ X k =0 (cid:16) (cid:17) k " , , , , , , k (cid:8)
43 + 246 k + 414 k (cid:9) . D3 . For a = c = q and e = q / , we find the corresponding identityΓ q (cid:18) (cid:19) Γ q (cid:18) (cid:19) = ∞ X k =0 q ( k +1)(2 k +1 / − q ( q ; q ) k ( q ; q ) k +1 ( q ; q ) k ( q ; q ) k ( q ; q ) k +1 × (cid:26) − (1 − q − k − )(1 − q k +1 )(1 − q k + ) + q k + (1 − q k + ) (1 − q k + )(1 − q k +2 ) (cid:27) which results in a q -analogue of the classical series40 π √ ∞ X k =0 (cid:16) (cid:17) k " , , , , , , k (cid:8)
214 + 591 k + 414 k (cid:9) . q -Pfaff–Saalsch¨utz theorem (5) as φ " q − n , q ⌊ n ⌋ a, q ⌊ n ⌋ cae, q −⌊ n ⌋ c/e (cid:12)(cid:12)(cid:12) q ; q = " q −⌊ n ⌋ e, q −⌊ n ⌋ ae/cae, q −⌊ n ⌋ e/c (cid:12)(cid:12)(cid:12) q n we obtain two further q -series identities. D4 . For a = q / and c = e = q / , the corresponding identity reads as1Γ q ( )Γ q ( ) = ∞ X k =0 − q k + − q ( q ; q ) k ( q ; q ) k h q , q ; q i k +1 ( q ; q ) k ( q ; q ) k +1 q k +2 k × (cid:26) − (1 − q − k )(1 − q k +1 ) (1 − q k + )(1 − q k + )(1 − q k + ) − q k + (1 − q k + )(1 − q k + ) (1 − q k + )(1 − q k +1 )(1 − q k +2 ) (1 − q k + ) (cid:27) which provides a q-analogue of the series729 √ π = ∞ X k =0 (cid:16) (cid:17) k " , , , , , , k (cid:8)
100 + 1521 k + 2610 k (cid:9) . D5 . For a = c = q and e = q / , the corresponding identity can be stated asΓ q (cid:18) (cid:19) = ∞ X k =0 q k + k q ( q ; q ) k ( q ; q ) k ( q ; q ) k ( q ; q ) k ( q ; q ) k × (cid:26) q k + (1 − q k + )(1 − q k +2 )(1 − q k +1 )(1 − q k + ) − q k + (1 − q k + )(1 − q k +1 )(1 − q k + )(1 − q k +3 )(1 − q k +1 )(1 − q k + )(1 − q k +2 )(1 − q k + ) (cid:27) . By carrying on the same procedure as done for C2 , we can confirm that the lastseries is, in fact, the bisection series of the following oneΓ q (cid:18) (cid:19) = ∞ X k =0 q k (3+5 k ) ( q ; q ) k ( q ; q ) k ( q ; q ) k q + k − q k − q k − q . This is in turn the q -analogue of the classical series (cf. Zhang [22, Example 8]): π = ∞ X k =0 (cid:16) (cid:17) k " , , k (cid:0) k (cid:1) = ∞ X k =0 k k (cid:0) k +2 k +1 (cid:1) ( k + 1)(2 k + 1) . Conclusive Comments.
We have shown that the inversion technique is efficientfor obtaining q -series identities whose limiting cases result in π -involved series. Theexamples presented in this paper are far from exhaustive. For instance, if we startwith the quadruplicate form of the q -Pfaff–Saalsch¨utz theorem (5) φ " q − n , q ⌊ n ⌋ a, q ⌊ n ⌋ cae, q −⌊ n ⌋ c/e (cid:12)(cid:12)(cid:12) q ; q = " q −⌊ n ⌋ e, q −⌊ n ⌋ ae/cae, q −⌊ n ⌋ e/c (cid:12)(cid:12)(cid:12) q n , then its dual series will give rise to the bisection series of the following q -series1Γ q ( )Γ q ( ) = ∞ X k =0 ( − k ( q ; q ) k ( q ; q ) k ( q ; q ) k ( q ; q ) k q k × (cid:26) − q + k − q − q + k (1 − q + k )(1 − q )(1 + q + k ) (1 + q + k ) (cid:27) which turns out to be a q -analogue of the elegant series for √ π with convergencerate “ − ” discovered by Guillera [12]:32 √ π = ∞ X k =0 (cid:16) − (cid:17) k " , , , , k (cid:8)
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School of StatisticsQufu Normal UniversityQufu (Shandong), P. R. China
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