Two generalizedq-exponential operators and their applications

Transcription

1 Li and an Advances in Difference quations (2016) 2016:53 DOI /s R S A R C H Open Access o generalizedq-exponential operators and their applications Nadia Na Li * and Wei an * Correspondence: lina3718@163com School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, , PR China Abstract In this paper, e construct to generalized q-exponential operators ith three parameters and obtain some operator identities As applications, e give to formal extensions of the q-gauss sum An extension of the q-chu-vandermonde sums is also obtained by the operator technique In addition, e also deduce a formal extension of the Askey-Wilson integral and a formal extension of Sears to-term summation formula Meanhile, some curious q-series identities are derived MSC: Primary 33D15; secondary 05A15 Keyords: q-series; q-exponential operator; q-integral; q-chu-vandermonde sums; Askey-Wilson integral 1 Introduction and motivation Folloing 1edefinetheq-shifted factorial by n 1 ( (a; q) 0 1, (a; q) n 1 aq k ) (, (a; q) 1 aq k ), and (a 1, a 2,,a m ; q) n (a 1 ; q) n (a 2 ; q) n (a m ; q) n,herem is a positive integer and n is a nonnegative integer or For a complex number α, e shall also use the notation (a; q) α (a; q) / ( q α a; q ) he q-binomial coefficients, or the Gauss coefficients, are given by n k (q; q) k k he basic hypergeometric series r φ s is defined by a 1, a 2,,a r rφ s q; z b 1, b 2,,b s (a 1, a 2,,a r ; q) k (q, b 1, b 2,,b r ; q) k ( 1) k q (k 2) 1+s r z k he usual q-differential operator, or q-derivative are defined by D x { f (x) f (x) f (qx) x { f (q 1 x) f (x), θ x f (x), q 1 x 2016 Li and an his article is distributed under the terms of the Creative Commons Attribution 40 International License ( hich permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes ere made

2 Li and an Advances in Difference quations (2016) 2016:53 Page 2 of 14 and their Leibniz rules are D n x θ n x { n f (x)g(x) q k(k n) k { f (x)g(x) D k x { { ( f (x) D n k x g xq k ), (1) n θ k { { ( x f (x) θ n k x g xq k ), (2) k respectively Based on this, Chen and Liu 2, 3 constructed the folloing augmentation operator, hich is of great significance for deriving identities from its special cases (cf 4 7): (bd x ) (bd x ) (bd x ) n, q (n 2) (bθx ) n Later, Chen andgu 8 defined the Cauchy augmentation operator, (a, bd x ) (a, bd x ) (a; q) n (bd x ) n, (a; q) n ( bθ x ) n Fang 9 and Zhang and Yang 10 considered the folloing finite generalized q-exponential operators ith to parameters: q N, N v q N, N q; tθ x v (q N, ; q) n (q, v; q) n (td x ) n, (q N, ; q) n (q, v; q) n (tθ x ) n In resent years, some authors applied the aforementioned q-exponential operators to give a system ay ofobtaining q-series formulas (cf 9 15) Inspired by their ork, in this paper, e construct to generalized q-exponential operator ith three parameters and give some operator identities to further investigate the applications of the operator technique Our paper is organized as follos he next section is devoted to the construction of to generalized q-exponentialoperatoriththreeparametersandeestablishsomeoperator identities he rest of this paper is mainly concerned ith applications of these operator identities In detail, e ill give to formal extensions of the q-gauss sum An extension of the q-chu-vandermonde sums are also obtained by the operator technique In addition, e also deduce a formal extension of the Askey-Wilson integral and a formal extension of Sears to-term summation formula Meanhile, some curious q-series identities are derived

3 Li and an Advances in Difference quations (2016) 2016:53 Page 3 of 14 2 o generalized q-exponential operators and their operator identities In this section, e first of all establish the folloing to generalized q-exponential operators ith three parameters: q; tθ x (; q) n (q, ; q) n (td x ) n, (3) (; q) n (q, ; q) n (tθ x ) n (4) Clearly, Zhang and Yang s 10 finite q-exponential operators ith to parameters can be considered as special cases of the operator (3) and(4) foru q N or v q N Basedon the definitions for D x and θ x and their Leibniz rules (1) and(2), e can easily verify the folloing explicit formulas by induction Lemma 1 For a nonnegative integer k, e have θx k { (xt; q) ( t) k (xt; q), { D k 1 t k x, (xt; q) (xt; q) { (xv; θx k q) t k (xv; q) q (k 2) (v/t; q)k, (xt; q) (xtq k ; q) { (xv; D k q) x t k (xvq k ; q) (v/t; q) k, (xt; q) (xt; q) θx k { x n ( 1) k x n k q k (q n ; q) k if 0 k n, 0 if k > n, D k x{ x n x n k (q n k+1 ; q) k if 0 k n, 0 if k > n heorem 2 { (xa; q) (xb, xc; q) (xa; q) (xb, xc; q) (; q) n+k (; q) n+k (a/b, xc; q) k (q, xa; q) k (bt) k (tc) n Proof By means of the definition (3) of q; td x operator and the Leibniz rule (1) of the q-derivative operator, e have { (xa; q) (xb, xc; q) (; q) n t n (q, ; q) n n q k(k n) D k x k { (xa; q) { D n k x 1 (q k xc; q)

4 Li and an Advances in Difference quations (2016) 2016:53 Page 4 of 14 In vie of Lemma 1, e can evaluate the right-hand side of the last expression in the folloing form: (; q) n t n (q, ; q) n (xa; q) (xb, xc; q) n q k(k n) b k (q k xa; q) (a/b; q) k k (; q) n (ct) n (q, ; q) n (q k c) n k (q k xc; q) n (a/b, xc; q)k (b/c) k k (xa; q) k Interchanging the summation order and changing by n n + k on the inner summation index, e get the desired result We remark that hen u q N and t q/c, the last theorem reduces to Zhang and Yang s conclusion 10 Similar to the procedure of proving heorem 2, e establish the folloing operator identity, hich leads to Zhang and Yang s conclusion 10 henu q N or v q N heorem 3 {(xa, xc; q) q; tθ x (xa, xc; q) (; q) n+k (; q) n+k (a/b, q/xc; q) k (q, q/xb; q) k ( 1) n (tc) n+k q (k 2) nk Letting c 0,thelastsumithrespectton in heorem 2 and heorem 3 vanishes because 0 n isequaltozerohenn > 0, therefore e get the folloing identities, respectively Corollary 4 {(xa; q) {(xa; q) q; tθ x (xa; q) 3φ 2 (xa; q) 3φ 2, a/b, xa, a/b, q/xb q; tb, (5) q; qt/x (6) For the symmetry of b and c on both sides of heorem 2,ehave (; q) n+k (a/b, xc; q) k (bt) k (tc) n (; q) n+k (q, xa; q) k (; q) n+k (a/c, xb; q) k (; q) n+k (q, xa; q) k (ct) k (tb) n Similarly, by means of the symmetry of a and c,egetfromheorem3 (; q) n+k (tc) n+k (a/b, q/xc; q) k ( 1) n q (k 2) nk (; q) n+k (q, q/xb; q) k (; q) n+k (; q) n+k (ta) n+k (b/c, q/xa; q) k (q, q/xb; q) k ( 1) n q (k 2) nk

5 Li and an Advances in Difference quations (2016) 2016:53 Page 5 of 14 Letting n + k m, and extracting the coefficients of (u,v;q) m (;q) m t m from both members of the last to identities, respectively, e get the folloing curious transformation formulas xample 1 q m, a/b, xc 3φ 1 q; q m b/c (b/c) m3φ 1 xa q m, a/b, q/xc 3φ 2 q; q (a/c) m3φ 2 0, q/xb q m, a/c, xb xa q m, c/b, q/xa 0, q/xb q; q m c/b, q; q (/u,/v;q) Letting t /uvc,thesumithrespectton in heorem 3 reduces to by (q k,q k /uv;q) means ofthe q-gauss sum a, b 2φ 1 q; c/ab c (c/a, c/b; q) (c, c/ab; q), (7) hich results consequently in the folloing operator identity Corollary 5 {(xa, q; xc; uvc θ q) x (xa, xc, /u, /b; q) a/b, q/xc, 4φ 3 q; q (xb,, /uv; q) 0, q/xb, quv/ Recalling the q-chu-vandermonde sums 1, II6, q n, x 2φ 1 q; q y x n (y/x; q) n (y; q) n, (8) and substituting x xa into the above equation, e derive (q n ; q) k q k (q, y; q) k (xaq k, xb; q) (xa)n (y/xa; q) n (y; q) n (xa, xb; q) (9) hen applying the operator q; td x to both sides of (9) ith respect to the variable x,itistrivialtoseethat (q n ; q) k q k { (q, y; q) k (xaq k, xb; q) an {x n (y/xa; q) n (y; q) n (xa, xb; q) valuating the left-hand side of the last expression in terms of heorem 2,eshofrom the last equation the folloing identity

6 Li and an Advances in Difference quations (2016) 2016:53 Page 6 of 14 heorem 6 For a nonnegative integer n, e have {Pn (x, y/a) (xa, xb; q) (y; q) n a n (xa, xb; q) (q n, xa; q) k q k (; q) i+j (c/b, q k xa; q) j (tb) j( q k ta ) i, (q, y; q) k (q; q) i (; q) i+j (q, xc; q) j i,j 0 here P n (a, b)a n (b/a; q) n When y 0,heorem6 reduces to the folloing corollary Corollary 7 For a nonnegative integer n, e have { x n (xa, xb; q) 1 a n (xa, xb; q) (q n, xa; q) k q k (; q) i+j (c/b, q k xa; q) j (tb) j( q k ta ) i (q; q) k (q; q) i,j 0 i (; q) i+j (q, xc; q) j Analogously, substituting x xa into (7), e can reformulate it as (q n ; q) k q k (q, y; q) k (xb, xc; q) (xaq k ; q) (xa)n (y/xa; q) n (xb, xc; q) (y; q) n (xa; q) Applying the operator q; tθ x to both sides of the last expression ith respect to the variable x, e confirm that (q n ; q) k q k {(xb, xc; q) q; tθ x (q, y; q) k (xaq k ; q) an {x n (y/xa; q) n (xb, xc; q) q; tθ x (y; q) n (xa; q) Reriting the left-hand side of the last expression by means of heorem 3,eobtainthe folloing identity heorem 8 For a nonnegative integer n, e have {Pn (x, y/a)(xb, xc; q) q; tθ x (xa; q) (y; q) n a n (xc, xb; q) (xa; q) (q n, xa; q) k q k (q, y; q) k i,j 0 (; q) i+j (q; q) i (; q) i+j (bt) i+j (cq k /a, q/xb; q) j (q, q 1 k /xa; q) j ( 1) i q (j 2) ij, here P n (a, b)a n (b/a; q) n

7 Li and an Advances in Difference quations (2016) 2016:53 Page 7 of 14 When b q/x, the inner sum ith respect to j vanishes, and then changing the consequent summation order, in vie of the q-chu-vandermonde sum (8), e get from heorem 8 {Pn (x, y/a) q; tθ x (xa; q) xn (y/xa; q) n 2φ 1 q; qt/x (xa; q) For t x/uvq, the right-hand side of the last identity can be evaluated by (7) We therefore establish the folloing operator identity Corollary 9 q; x {Pn uvq θ (x, y/a) x (xa; q) xn (y/xa; q) n (xc, /u, /v; q) (xa,, /uv; q) 3 Some applications hroughout this section, e address the applications of the theorems and corollaries obtained in the previous section 31 o formal generalizations of the q-gauss sum Recalling the q-binomial theorem 1, II3, (a; q) n x n (xa; q) (x; q), (10) multiplying (xc;q) (xb;q) to both sides of the last identity, and applying q; td x to both sides of the consequent identity ith respect to the parameter x,ehave (a; q) n { x n (xa, xb; q) { (xc; q) (x, xb; q) By means of heorem 2 and Corollary 7, e have the folloing theorem heorem 10 (a; q) n a n (xa; q) (x; q) i,j 0 (q n, xa; q) k q k (q; q) k i,j 0 (; q) i+j (q; q) i (; q) i+j (c, xb; q) j (q, xc; q) j t j (tb) i (; q) i+j (c/b, q k xa; q) j (q; q) i (; q) i+j (q, xc; q) j (bt) j( q k ta ) i If c 1andt /uvb,thesumithrespecttoj on the right-hand side of the last theorem vanished and the sum ith respect to i can be evaluated by the q-gauss sum (7) We find consequently the folloing curious summation formula here the parameter b only exists in the left-hand side

8 Li and an Advances in Difference quations (2016) 2016:53 Page 8 of 14 Corollary 11 (a; q) n a n i,j 0 (q n, xa; q) k q k (q; q) k (; q) i+j (q; q) i (; q) i+j (1/b, q k xa; q) j (q, x; q) j (xa, /u, /v; q) (x,, /uv; q) ( uv ) j ( q k ) a i uvb Further, if e specify ith a 1 in the last corollary, e find the folloing curious summation formula, hich reduces to the q-gauss sum (7)henb 1 Corollary 12 i,j 0 ( ) (; q) i+j (1/b; q) j j ( ) i (/u, /v; q) (q; q) i (; q) i+j (q; q) j uv uvb (, /uv; q) Letting i + j m, and then extracting the coefficients of (u,v;q) m (;q) m t m from to members of heorem 10, eget thefolloingresult Corollary 13 (a; q) n a m n m j0 b m (xa; q) (x; q) 3φ 1 (q n, xb; q) k q (1+m)k (q; q) k (q m, c/b, q k xa; q) j ( 1) j q (j 2) ( bq m k /a ) j (q, xc; q) j q m, c, xb xc q; q m /b When a 1, the last sums ith respect to n and k vanish because of the q-shift factorial (1; q) n isequaltozerohenn > 0 hen e get the folloing transformation formula xample 2 q m, c/b, x 3φ 1 q; q m b b m3φ q m, c, xb 1 xc xc q; q m /b Similarly, multiplying (xb;q) (xc;q) to both sides of equation (10) and applying q; tθ x to both sides of the consequent identity ith respect to the parameter x,e have (a; q) n {x n (x, xb; q) q; tθ x {(xa, xb; q) q; tθ x (11)

9 Li and an Advances in Difference quations (2016) 2016:53 Page 9 of 14 On the one hand, letting y 0 and substituting ith a c, c 1inheorem8,ehave {x n (x, xb; q) q; tθ x 1 c n (xb, x; q) i,j 0 (q n, xc; q) k q k (q; q) k (; q) i+j (q; q) i (; q) i+j (bt) i+j (q k /c, q/xb; q) j (q, q 1 k /xc; q) j ( 1) i q (j 2) ij On the other hand, in vie of heorem 3,eget {(xa, xb; q) q; tθ x (xa, xb; q) (; q) n+k (; q) n+k (tb) n+k (a/c, q/xb; q) k (q, q/xc; q) k ( 1) n q (k 2) nk Combining the last to expressions ith (11), e get the folloing theorem heorem 14 (a; q) n 1 (q n, xc; q) k q k c n (q; q) k (xa; q) (x; q) i,j 0 (; q) i+j (bt) i+j (q; q) i (; q) i+j (q k /c, q/xb; q) j (q, q 1 k /xc; q) j ( 1) i q (j 2) ij (; q) n+k (; q) n+k (tb) n+k (a/c, q/xb; q) k (q, q/xc; q) k ( 1) n q (k 2) nk If a c,thesumithrespecttok on the right-hand side of the above theorem vanishes Further setting t /uvb in the consequent expression, e get the folloing summation formula, hich returns to the q-gauss sum (7)hena 1 Corollary 15 (a; q) n 1 (q n, xa; q) k q k (; q) i+j (q k /a, q/xb; q) j a n (q; q) k (q; q) i (; q) i+j (q, q 1 k /xa; q) j ( /uv) i+j ( 1) i q (j 2) ij (xa, /u, /v; q) (x,, /uv; q) i,j 0 32 A formal generalization of q-exponential function heorem 16 q (n 2) x n (q n (, ; q) k q ) k ( x, /u, /v; q) (q, ; q) k xuv (, /uv; q) Proof Recalling the q-exponential function 1, II2 q (n 2) x n ( x; q)

10 Li and an Advances in Difference quations (2016) 2016:53 Page 10 of 14 and then applying q; uv θ x to the both sides of the last identity, e have q (n 2) q; {x x uv θ n q; {( x; x uv θ q) (12) On the one hand, letting b 0,a 1andt /uv in (6), e can rite the right-hand side of (12)as q; {( x; uv θ ( x, /u, /v; q) x q) (, /uv; q) by means ofthe q-gauss sum (7) On the other hand, in vie of the definition (4) andtheequationofθ k x {xn obtained in Lemma 1,ecaneasilyverifythat {x n q; /uvθ x x n q n, 3φ 2 q; q/uvx,0 Inserting the last to expressions into (12), e obtain the desired result hich reduces to the q-exponential function hen u 1orv 1 33 Ageneralizationof q-chu-vandermonde sums In vie of the fact that and (a; q) n (a; q) (q n a; q) (q/a; q) n ( a) n q (n+1 2 ) (q n a; q) (a; q), e can reformulate the q-chu-vandermonde sums (8)as (q n ; q) k q k (q, c; q) k 1 ( 1)n c n q ( n 2 ) (xq k ; q) (c; q) n (q 1 n x/c; q) (x, qx/c) Applying q; td x to both sides of the last expression, e have (q n ; q) k q k (q, c; q) k ( 1)n c n q (n 2) (c; q) n { 1 (xq k ; q) {(q 1 n x/c; q) (x, qx/c) On the one hand, specifying ith a 0andb q k,ecanrestate(5)as { 1 (xq k ; q) 1 2φ (q k 1 x; q) q; q k t

11 Li and an Advances in Difference quations (2016) 2016:53 Page 11 of 14 On the other hand, in vie of heorem 2,itisnothardtoderive {(q 1 n x/c; q) (x, qx/c) (q1 n x/c; q) (x, qx/c; q) i,k 0 (; q) i+k (q; q) i (; q) i+k (q 1 n /c, qx/c; q) k (q, q 1 n x/c; q) k t i+k (q/c) i Combining the last three expression, e get the folloing result, hich is an extension of the q-chu-vandermonde sums (8) We remark that it leads to Zhang and Yang s result (10, heorem 31), hen u or v equals q N and t c heorem 17 (q n, x; q) k q k (q, c; q) k 2φ 1 xn (c/x; q) n (c; q) n i,k 0 q; q k t (; q) i+k (q; q) i (; q) i+k (q 1 n /c, qx/c; q) k (q, q 1 n x/c; q) k t i+k (q/c) i Supposing n 0andt /uv, e can reduce the previous theorem as follos Corollary 18 i,k 0 (; q) i+k (q/c; q) k (q; q) i (; q) i+k (q; q) k (/uv) i+k (q/c) i (/u, /v; q) (, /uv; q) Letting i + k m, and then extracting the coefficients of (u,v;q) m (;q) m t m from both members of heorem 17, e get the folloing curious transformation formula, hich results in the q-chu-vandermonde sums (8)henm 0 Corollary 19 q n, x 2φ 1 q; q m+1 c (c/x; q) n (c; q) n x n (q/c) m 3φ 1 q m, q 1 n /c, qx/c q 1 n x/c 34 A formal generalization of Sears formula Recalling the Sears to-term summation formula 1, q (21018) q; q m 1 c d c (qt/c, qt/d, abcdet; q) (at, bt, et; q) e can restate it as d q t d(1 q)(q, qd/c, c/d, abcd, acde; q) (ac, ad, bc, bd, ce, de; q), (13) d c (qt/c, qt/d; q) (bt, et; q) (abcdet; q) (at, acde; q) d q t d(1 q)(q, qd/c, c/d; q) (bd, ce, de; q) (abcd; q) (ac, ad; q)

12 Li and an Advances in Difference quations (2016) 2016:53 Page 12 of 14 Further applying q; sd x to both sides of the above identity ith respect to the variable a, e rite don immediately that d c (qt/c, qt/d; q) (bt, et; q) d(1 q)(q, qd/c, c/d; q) (bd, ce, de; q) { (abcdet; q) q; sd x d q t (at, acde; q) { (abcd; q) q; sd x (ac, ad; q) Applying heorem 2 to both sides of the last identity, e therefore establish the folloing transformation formula heorem 20 d c (qt/c, qt/d, abcdet; q) (at, bt, et; q) n, d(1 q)(q, qd/c, c/d, abcd, acde; q) (ac, ad, bc, bd, ce, de; q) (; q) n+k s n+k (at, bt; q) k (cde) k t n d q t (; q) n+k (q, abcdet; q) k n, (; q) n+k s n+k (; q) n+k (bd, ad; q) k (q, abcd; q) k c k d n Letting n + k m, and then extracting the coefficients of (u,v;q) m (;q) m s m from both members of the last theorem, e derive the folloing result hich leads to the Sears to-term summation formula (13)henm 0 Corollary 21 d c t m (qt/c, qt/d, abcdet; q) (at, bt, et; q) 3φ 1 q m, at, bt abcdet dm+1 (1 q)(q, qd/c, c/d, abcd, acde; q) (ac, ad, bc, bd, ce, de; q) 3φ 1 35 A formal generalization of Askey-Wilson integral Recalling the Askey-Wilson integral 13 π 0 e express it as π 0 h(cos 2θ;1)dθ h(cos θ; b, c, d, x) 2π (q, bc, bd, cd; q) q; q m cde/t d q t q m, ad, bd abcd q; q m c/d (xbcd; q) (xb, xc, xd; q), (14) h(cos 2θ;1) 2π (xbcd; q) dθ h(cos θ; b, c, d) (xe iθ, xe iθ ; q) (q, bc, bd, cd; q) (xc, xd; q) Applying q; td x to both sides of the above identity ith respect to the variable x, e get π 0 h(cos 2θ;1) h(cos θ; b, c, d) 2π (q, bc, bd, cd; q) { (xe iθ, xe iθ ; q) { (xbcd; q) (xc, xd; q) dθ (15) (16)

13 Li and an Advances in Difference quations (2016) 2016:53 Page 13 of 14 By means of heorem 2,eget,respectively, and { (xe iθ, xe iθ ; q) (xe iθ, xe iθ ; q) { (xbcd; q) (xc, xd; q) (xbcd; q) (xc, xd; q) (; q) n+k (be iθ, xe iθ ; q) k ( te iθ ) k( te iθ ) n (; q) n+k (q, xb; q) k (; q) n+k (; q) n+k (bd, xd; q) k (q, xbcd; q) k (tc) k (td) n Substituting the last to expressions into the identity (15), e have the folloing extension of the Askey-Wilson integral heorem 22 For max{ b, c, d, x <1,e have π 0 h(cos 2θ;1) h(cos θ; b, c, d, x) (; q) n+k t n+k (be iθ, xe iθ ; q) k e (k n)iθ dθ (; q) n+k (q, xb; q) k 2π(xbcd; q) (q, bc, bd, cd, xb, xc, xd; q) (; q) n+k (; q) n+k (bd, xd; q) k (q, xbcd; q) k (tc) k (td) n Letting n + k m, and then extracting the coefficients of (u,v;q) m (;q) m t m from both members of the last theorem, e thus give the folloing transformation formula, hich leads to the Askey-Wilson integral (14)henm 0 Corollary 23 π 0 h(cos 2θ;1) h(cos θ; b, c, d, x) m d m 2π(xbcd; q) (q, bc, bd, cd, xb, xc, xd; q) m (be iθ, xe iθ ; q) k e (2k m)iθ dθ k (xb; q) k m m (bd, xd; q)k (c/d) k k (xbcd; q) k Competing interests he authors declare that they have no competing interests Authors contributions All authors contributed equally to the riting of this paper All authors read and approved the final manuscript Acknoledgements his research as supported by the ducation Department of Henan Province (Grant No 15A110046) and the Research Innovation Project of Zhoukou Normal University (Grant No zknua201410) Received: 3 September 2015 Accepted: 4 January 2016

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