A note on fractional Askey--Wilson integrals
aa r X i v : . [ m a t h . C A ] J a n A NOTE ON FRACTIONAL ASKEY–WILSON INTEGRALS
JIAN CAO AND SAMA ARJIKA A bstract . In this paper, we generalize fractional q -integrals by the method of q -di ff erence equation. In addition,we deduce fractional Askey–Wilson integral, reversal type fractional Askey–Wilson integral and Ramanujan typefractional Askey–Wilson integral.
1. I ntroduction
The fractional q -calculus theories have been applied successfully in many fields, which is a very suitabletool in describing and solving a lot of problems in numerous sciences [23, 18], such as high-energy physics,system control, biomedical engineering and economics. Its theoretical and applied research has become a hotspot in the world. The treatment from the point view of the q -calculus can open new perspectives as it did, forexample, in optimal control problems [7, 8, 15, 27]. For more information, see details in [10, 11, 2, 3, 21, 29,30, 31, 32, 33, 24, 25, 26].In this paper, we follow the notations and terminology in [16] and suppose that 0 < q <
1. We first show alist of various definitions and notations in q -calculus which are useful to understand the subject of this paper.The basic hypergeometric series r φ s [16] r φ s (cid:20) a , a , . . . , a r b , b , . . . , b s ; q , z (cid:21) = ∞ X n = (cid:0) a , a , . . . , a r ; q (cid:1) n (cid:0) q , b , b , . . . , b s ; q (cid:1) n h ( − n q ( n ) i + s − r z n (1.1)converges absolutely for all z if r ≤ s and for | z | < r = s + r φ s are defined respectively by( a ; q ) = , [ a ] q : = − q a − q , ( a ; q ) n = n − Y k = (1 − aq k ) , ( a ; q ) ∞ = ∞ Y k = (1 − aq k ) (1.2)and ( a , a , . . . , a m ; q ) n = ( a ; q ) n ( a ; q ) n · · · ( a m ; q ) n , where m ∈ N : = { , , , · · · } and n ∈ N : = N ∪ { } . The q -gamma function is defined by [16] Γ q ( x ) = ( q ; q ) ∞ ( q x ; q ) ∞ (1 − q ) − x , x ∈ R \{ , − , − , . . . } . (1.3)The Thomae–Jackson q -integral is defined by [16, 17, 28] Z ba f ( x ) d q x = (1 − q ) ∞ X n = h b f ( bq n ) − a f ( aq n ) i q n . (1.4)The Riemann–Liouville fractional q -integral operator is introduced in [1] (cid:16) I α q f (cid:17) ( x ) = x α − Γ q ( α ) Z x (cid:0) qt / x ; q (cid:1) α − f ( t ) d q t . (1.5) Key words and phrases.
Fractional q -integral; Askey–Wilson integral; q -di ff erence equation. Department of Mathematics, Hangzhou Normal University, Hangzhou City, Zhejiang Province, 311121, China. Department ofMathematics and Informatics, University of Agadez, Niger.Email: [email protected], [email protected]
Mathematics Subject Classification .05A30, 11B65, 33D15, 33D45, 33D60, 39A13, 39B32. AND SAMA ARJIKA The generalized Riemann–Liouville fractional q -integral operator is given by [25] (cid:16) I α q , a f (cid:17) ( x ) = x α − Γ q ( α ) Z xa (cid:0) qt / x ; q (cid:1) α − f ( t ) d q t , α ∈ R + . (1.6)In fact, we rewrite fractional q -integral (1.6) equivalently as follows by (1.4) (cid:16) I α q , a f (cid:17) ( x ) = x α − (1 − q ) Γ q ( α ) ∞ X n = h x (cid:0) q n + ; q (cid:1) α − f (cid:0) xq n (cid:1) − a (cid:0) aq n + / x ; q (cid:1) α − f (cid:0) aq n (cid:1)i q n . In paper [13], author built the relations between the following fractional q -integrals and certain generatingfunctions for q -polynomials. Proposition 1 ([13, Theorem 3]) . For α ∈ R + and < a < x < , if max {| at | , | az |} < , we haveI α q , a (cid:26) ( bxz , xt ; q ) ∞ ( xs , xz ; q ) ∞ (cid:27) = (1 − q ) α ( abz , at ; q ) ∞ ( as , az ; q ) ∞ ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k φ " q − k , as , azat , abz ; q , q . (1.7)In this paper, we generalize fractional q -integrals and give applications of fractional Askey–Wilson integralsas follows. Theorem 2.
For α ∈ R + and < a < x < , if max {| at | , | az | , | aru |} < , we haveI α q , a (cid:26) ( bxz , xt , xru ; q ) ∞ ( xs , xz , xu ; q ) ∞ (cid:27) = (1 − q ) α ( abz , at , aru ; q ) ∞ ( as , az , au ; q ) ∞ ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k φ " q − k , as , az , auabz , at , aru ; q , q . (1.8) Remark 3.
For u = in Theorem 2 , equation (1.8) reduces to (1.7) . Before the proof of Theorem 2, the following lemmas are necessary.
Lemma 4 ([22, Proposition 1.2]) . Let f ( a , b , c ) be a three-variable analytic function in a neighborhood of ( a , b , c ) = (0 , , ∈ C . If f ( a , b , c ) satisfies the di ff erence equation ( c − b ) f ( a , b , c ) = ab f ( a , bq , cq ) − b f ( a , b , cq ) + ( c − ab ) f ( a , bq , c ) , (1.9) then we have f ( a , b , c ) = T ( a , bD c ) { f ( a , , c ) } , (1.10) where T ( a , bD c ) = ∞ X n = ( a ; q ) n ( q ; q ) n ( bD c ) n , D c { f ( c ) } = f ( c ) − f ( cq ) c . (1.11) Lemma 5.
For max {| as | , | az | , | au |} < , we have ( s − u ) ( abz , at , aru ; q ) ∞ ( as , az , au ) ∞ = ur ( abz , at , aruq ; q ) ∞ ( asq , az , auq ) ∞ − u ( abz , at , aru ; q ) ∞ ( asq , az , au ) ∞ + ( s − ur ) ( abz , at , aruq ; q ) ∞ ( as , az , auq ) ∞ . (1.12) Proof of Lemma 5.
The right-hand side (RHS) of equation (1.12) equals ur (1 − xs )(1 − xu )1 − xur ( abz , at , aru ; q ) ∞ ( as , az , au ) ∞ − u (1 − xs ) ( abz , at , aru ; q ) ∞ ( as , az , au ) ∞ + ( s − ur ) 1 − xu − xur ( abz , at , aru ; q ) ∞ ( as , az , au ) ∞ = ( s − u ) ( abz , at , aru ; q ) ∞ ( as , az , au ) ∞ , (1.13)which is equals to the left-hand side (LHS) of the equation (1.12). The proof is complete. Lemma 6 ([14, Eq. (2.3)]) . For max {| bt | , | ct |} < , we haveT ( a , bD c ) ( ct ; q ) ∞ ) = ( abt ; q ) ∞ ( bt , ct ; q ) ∞ . (1.14) NOTE ON FRACTIONAL ASKEY–WILSON INTEGRALS 3
Proof of Theorem 2.
Denoting the RHS of the equation (1.8) by f ( r , u , s ), and rewriting f ( r , u , s ) equivalentlyby f ( r , u , s ) = ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k k X j = ( q − k ; q ) j q j ( q ; q ) j · (1 − q ) α ( abzq j , atq j , aruq j ; q ) ∞ ( asq j , azq j , auq j ; q ) ∞ , we check that f ( r , u , s ) satisfies the equation (1.9) by Lemma 5, then we have f ( r , u , s ) = T ( r , uD s ) { f ( r , , s ) } = T ( r , uD s ) ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k k X j = ( q − k ; q ) j q j ( q ; q ) j · (1 − q ) α ( abzq j , atq j ; q ) ∞ ( asq j , azq j ; q ) ∞ = T ( r , uD s ) ( I α q , a ( ( xbz , xt ; q ) ∞ ( xs , xz ; q ) ∞ )) = I α q , a ( T ( r , uD s ) ( ( xbz , xt ; q ) ∞ ( xs , xz ; q ) ∞ )) = I α q , a ( ( xbz , xt ; q ) ∞ ( xz ; q ) ∞ · T ( r , uD s ) ( xs ; q ) ∞ )) , which equals the LHS of the equation (1.8) by Lemma 6. The proof is complete.The rest of the paper is organized as follows. In Section 2, we give the fractional Askey–Wilson integral. InSection 3, we obtain the reversal type fractional Askey–Wilson integral. In Section 4, we get the Ramanujantype fractional Askey–Wilson integral.2. A generalization of A skey –W ilson integrals In 1985, Askey and Wilson gave the famous Askey–Wilson integral, which greatly promoted the researchand development of orthogonal polynomials. Chen and Gu [14], Liu [19] and Cao [9] et al have promotedAskey–Wilson integral by di ff erent methods. For more information, see details in [4, 9, 14, 19].In this section, we use the fractional q -integrals to generalize Askey–Wilson integrals. Proposition 7 ([4, Theorem 2.1]) . If max {| a | , | b | , | c | , | d |} < , we have Z π h (cos 2 θ ; 1) h (cos θ ; a , b , c , d ) d θ = π ( abcd ; q ) ∞ ( q , ab , ac , ad , bc , bd , cd ; q ) ∞ , (2.1) where h (cos θ ; a ) = ( ae i θ , ae − i θ ; q ) ∞ , h (cos θ ; a , a , · · · , a m ) = h (cos θ ; a ) h (cos θ ; a ) · · · h (cos θ ; a m ) . Theorem 8.
For α ∈ R + , if max {| a | , | b | , | c | , | d |} < , we have Z π h (cos 2 θ ; 1) h (cos θ ; a , b , c , d ) ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k φ " q − k , abcd , ae i θ , ae − i θ ab , ac , ad ; q , q d θ = π ( abcd ; q ) ∞ ( q , ab , ac , ad , bc , bd , cd ; q ) ∞ x α ( a / x ; q ) α ( q ; q ) α . (2.2) Corollary 9.
For α ∈ R + , if max {| a | , | b | , | c |} < , we have Z π h (cos 2 θ ; 1) h (cos θ ; a , b , c ) ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k φ " q − k , ae i θ , ae − i θ ab , ac ; q , q d θ = π ( q , ab , ac , bc ; q ) ∞ x α ( a / x ; q ) α ( q ; q ) α . (2.3) JIAN CAO AND SAMA ARJIKA Remark 10.
For d = in Theorem 8, equation (2.2) reduces to (2.3) .Proof of Theorem 8. The equation (2.2) can be rewrite equivalently by Z π h (cos 2 θ ; 1) h (cos θ ; b , c , d ) ( ab , ac , ad ; q ) ∞ ( ae i θ , ae − i θ , abcd ; q ) ∞ ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k × φ " q − k , abcd , ae i θ , ae − i θ ab , ac , ad ; q , q d θ = π ( q , bc , bd , cd ; q ) ∞ x α ( a / x ; q ) α ( q ; q ) α . (2.4)By using Theorem 2, the LHS of the equation (2.4) is equivalent to Z π h (cos 2 θ ; 1) h (cos θ ; b , c , d ) I α q , a (cid:26) ( xb , xc , xd ; q ) ∞ ( xe i θ , xe − i θ , xbcd ; q ) ∞ (cid:27) d θ = I α q , a (cid:26)Z π h (cos 2 θ ; 1)( xb , xc , xd ; q ) ∞ h (cos θ ; x , b , c , d )( xbcd ; q ) ∞ d θ (cid:27) = I α q , a (cid:26) π ( q , bc , bd , cd ; q ) ∞ (cid:27) = π ( q , bc , bd , cd ; q ) ∞ x α ( a / x ; q ) α ( q ; q ) α , which is the RHS of the equation (2.4). The proof is complete.3. A generalization of reversal type A skey –W ilson integrals In this section, we use the fractional q -integrals to expand reversal Askey–Wilson integrals. Proposition 11 (Reversal Askey–Wilson integral [5]) . For | qabcd | < , there holds Z ∞−∞ h ( i sinh x ; qa , qb , qc , qd ) h (cosh 2 x ; − q ) dx = ( q , qab , qac , qad , qbc , qbd , qcd ; q ) ∞ ( qabcd ; q ) ∞ log( q − ) , (3.1) where h ( i sinh α x ; t ) = ∞ Y k = (1 − iq k t sinh α x + q k t ) = ( ite α x , − ite − α x ; q ) ∞ . Theorem 12.
For α ∈ R + and | qabcd | < , we have Z ∞−∞ h ( i sinh t ; qa , qb , qc , qd ) h (cosh 2 t ; − q ) ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k φ " q − k , qab , qac , qadiaqe t , − iaqe − t , qabcd ; q , q dt = ( q , qab , qac , qad , qbc , qbd , qcd ; q ) ∞ ( qabcd ; q ) ∞ x α ( a / x ; q ) α ( q ; q ) α log( q − ) . (3.2) Corollary 13.
For α ∈ R + , we have Z ∞−∞ h ( i sinh t ; qa , qb , qc ) h (cosh 2 t ; − q ) ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k φ " q − k , qab , qaciaqe t , − iaqe − t ; q , q dt = ( q , qab , qac , qbc ; q ) ∞ x α ( a / x ; q ) α ( q ; q ) α log( q − ) . (3.3) Remark 14.
For d = in Theorem 12, equation (3.2) reduces to (3.1) .Proof of Theorem 12. The equation (3.2) can be rewrite Z ∞−∞ h ( i sinh t ; qb , qc , qd ) h (cosh 2 t ; − q ) ( iaqe t , − iaqe − t , qabcd ; q ) ∞ ( qab , qac , qad ; q ) ∞ ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k φ " q − k , qab , qac , qadiaqe t , − iaqe − t , qabcd ; q , q dt NOTE ON FRACTIONAL ASKEY–WILSON INTEGRALS 5 = ( q , qbc , qbd , qcd ; q ) ∞ x α ( a / x ; q ) α ( q ; q ) α log( q − ) , (3.4)By using Theorem 2, the LHS of the equation (3.4) equals Z ∞−∞ h ( i sinh t ; qb , qc , qd ) h (cosh 2 t ; − q ) I α q , a (cid:26) ( ixqe t , − ixqe − t , qxbcd ; q ) ∞ ( qxb , qxc , qxd ; q ) ∞ (cid:27) dt = I α q , a (cid:26)Z ∞−∞ h ( i sinh t ; qx , qb , qc , qd )( qxbcd ; q ) ∞ h (cosh 2 t ; − q )( qxb , qxc , qxd ; q ) ∞ dt (cid:27) = I α q , a (cid:26) ( q , qbc , qbd , qcd ; q ) ∞ log( q − ) (cid:27) = ( q , qbc , qbd , qcd ; q ) ∞ x α ( a / x ; q ) α ( q ; q ) α log( q − ) . which is the RHS of the equation (3.4). The proof is complete.4. A generalization of R amanujan type A skey –W ilson integrals In 1994, Atakishiyev discovered the Atakishiyev integral by Ramanujan’s method. After that, Wang [33]and Liu [20] et al conducted di ff erent generalization and proving lines for this integral.In this part, we get the generating functions of Atakishiyev integral by fractional q -integral. Proposition 15 (Atakishiyev integral [6]) . If α is a real number and q = exp ( − α ) , then we have Z ∞−∞ h ( i sinh α t ; a , b , c , d ) e − t cosh α tdt = √ π q − ( ab / q , ac / q , ad / q , bc / q , bd / q , cd / q ; q ) ∞ ( abcd / q ; q ) ∞ . (4.1) Theorem 16.
For α ∈ R + and (cid:12)(cid:12)(cid:12) abcd / q (cid:12)(cid:12)(cid:12) < , if α is a real number and q = exp ( − α ) , then we have Z ∞−∞ h ( i sinh α t ; a , b , c , d ) e − x cosh α t ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k φ " q − k , ab / q , ac / q , ad / qiae α t , − iae − α t , abcd / q ; q , q dt = √ π q − ( ab / q , ac / q , ad / q , bc / q , bd / q , cd / q ; q ) ∞ ( abcd / q ; q ) ∞ x α ( a / x ; q ) α ( q ; q ) α . (4.2) Corollary 17.
For α ∈ R + , if α is a real number and q = exp ( − α ) , then we have Z ∞−∞ h ( i sinh α t ; a , b , c ) e − x cosh α t ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k φ " q − k , ab / q , ac / qiae α t , − iae − α t ; q , q dt = √ π q − ( ab / q , ac / q , bc / q ; q ) ∞ x α ( a / x ; q ) α ( q ; q ) α . (4.3) Remark 18.
For d = in Theorem 16, equation (4.2) reduces to (4.3) .Proof of Theorem 16. The equation (4.2) can be rewrited Z ∞−∞ h ( i sinh α t ; b , c , d ) e − x cosh α t ( iae α t , − iae − α t , abcd / q ; q ) ∞ ( ab / q , ac / q , ad / q ; q ) ∞ ∞ X k = x α + k (cid:0) a / x ; q (cid:1) α + k a k ( q ; q ) α + k × φ " q − k , ab / q , ac / q , ad / qiae α t , − iae − α t , abcd / q ; q , q dt = √ π q − ( bc / q , bd / q , cd / q ; q ) ∞ x α ( a / x ; q ) α ( q ; q ) α . (4.4)By using Theorem 2, the LHS of the equation (4.4) is equal to Z ∞−∞ h ( i sinh α t ; b , c , d ) e − x cosh α t · I α q , a (cid:26) ( ixe α t , − ixe − α t , xbcd / q ; q ) ∞ ( xb / q , xc / q , xd / q ; q ) ∞ (cid:27) dt JIAN CAO AND SAMA ARJIKA = I α q , a (cid:26)Z ∞−∞ h ( i sinh α t ; x , b , c , d ) e − x cosh α t · ( xbcd / q ; q ) ∞ ( xb / q , xc / q , xd / q ; q ) ∞ dt (cid:27) = I α q , a (cid:26) √ π q − ( bc / q , bd / q , cd / q ; q ) ∞ (cid:27) = √ π q − ( bc / q , bd / q , cd / q ; q ) ∞ x α ( a / x ; q ) α ( q ; q ) α , which is the RHS of the equation (4.4). The proof is complete.A cknowledgments The author would like to thank the referees and editors for their many valuable comments and suggestions.This work was supported by the Zhejiang Provincial Natural Science Foundation of China (No. LY21A010019).R eferences [1] W.A. Al-Salam,
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