Fractional q-Leibniz Rule and Applications
This chapter includes analytic investigations on q-type Leibniz rules of q-Riemann–Liouville fractional operator introduced by Al-Salam and Verma in (Pac. J. Math. 60(2), 1–9, 1975). In this chapter, we provide a generalization of the Riemann–Liouville fr
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Fractional q-Leibniz Rule and Applications
Abstract This chapter includes analytic investigations on q-type Leibniz rules of q-Riemann–Liouville fractional operator introduced by Al-Salam and Verma in (Pac. J. Math. 60(2), 1–9, 1975). In this chapter, we provide a generalization of the Riemann–Liouville fractional q-Leibniz formula introduced by Agarwal in (Ganita 27(1–2), 25–32, 1976). Purohit (Kyungpook Math. J. 50(4), 473–482, 2010) introduced a Leibniz formula for Weyl q-fractional operator only when ˛ is an integer. In this respect, We extend Purohit’s result for any ˛ 2 R. We end the chapter with deriving some q-series and formulae by applying the fractional Leibniz formula mentioned and derived earlier in the chapter.
6.1 Leibniz Rule for Fractional Derivatives Studies of Leibniz rule for derivatives of arbitrary order started with the work of Liouville [186] who introduced the Leibniz rule D ˛ u.z/v.z/ D
1 X nD0
.˛ C 1/ D ˛n u.z/D n v.z/: .˛ n C 1/
(6.1)
While Liouville used Fourier expansions in obtaining (6.1), Gr¨unwald [120] and Letnikov [180] obtained (6.1) in a different technique. Other extensions and proofs could be found in the work of Watanabe [288], Post [242], Bassam [49] and Gaer– Rubel [110]. In Watanabe’s paper, the author derived the generalization D ˛ u.z/v.z/ D
1 X 1
.˛ C 1/ D ˛ n u.z/D Cn v.z/; (6.2) .˛ n C 1/ . C n C 1/
for any fixed . In [288], no precise domain of convergence is given [228, P. 659]. Osler in [228], see also [81, 229–231], determined the domain of convergence of (6.2) which includes (6.1) as the special case D 0. M.H. Annaby and Z.S. Mansour, q-Fractional Calculus and Equations, Lecture Notes in Mathematics 2056, DOI 10.1007/978-3-642-30898-7 6, © Springer-Verlag Berlin Heidelberg 2012
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The expansion (6.2) can be used to derive some series expansions of some functions for example [228, P. 659]. If we take u D zd a1 , v D zcb1 , ˛ D c a 1 and D c 1, we end with 1
X .a C n/ .b C n/ 2 .c C d a b 1/ csc a csc b D : .c a/ .c b/ .d a/ .d b/ .c C n/ .d C n/ 1 The identity F .a; bI c:z/ D
1 .c/ .b c C 1/ sin .b c/ X F .a; bI b nI z/ .1/n nŠ.b c n/ .b n/ nD0
which holds for Re b > 0 and Rez < 1=2 is obtained from (6.2) by calculating D bc zb1 .1 z/a by two different methods. One method is by taking U.z/ D zb1 ; V .z/ D .1 z/a ; and the other methods is by taking U.z/ D .1 z/a ;
V .z/ D zb1 :
6.2 Al-Salam–Verma Fractional q-Leibniz Formula In [20], Al-Salam and Verma derived formally a q-fractional Leibniz rule. They used the q-Taylor series f .z/ D
1 X .1 q/n .1/n q n.n1/=2 Dqn f .aq n / Œa zn ; .qI q/n nD0
(6.3)
which is slightly different from the q-Newton series f .z/ D
1 X nD0
Dqn f .a/
.1 q/n Œz an ; .qI q/n
(6.4)
introduced formally by Jackson in [159]. According to Jackson’s notations, Œy tn denotes the function n .y; t/ defined in (6.7) below for y; t 2 C and n 2 N0 . No
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