Abstract
In this paper we generalize the fractional q-Leibniz formula introduced by Agarwal in (Ganita 27(1-2):25-32, 1976) for the Riemann-Liouville fractional q-derivative. This extension is a q-version of a fractional Leibniz formula introduced by Osler in (SIAM J. Appl. Math. 18(3):658-674, 1970). We also introduce a generalization of the fractional q-Leibniz formula introduced by Purohit for the Weyl fractional q-difference operator in (Kyungpook Math. J. 50(4):473-482, 2010). Applications are included.
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1 q-notions and notations
Let q be a positive number, . In the following, we follow the notations and notions of q-hypergeometric functions, the q-gamma function , Jackson q-exponential functions , and the q-shifted factorial as in [1, 2]. By a q-geometric set A, we mean a set that satisfies if , then . Let f be a function defined on a q-geometric set A. The q-difference operator is defined by
The n th q-derivative, , can be represented by its values at the points through the identity
for every x in . After some straightforward manipulations, formula (2) can be written as
Moreover, formula (2) can be inverted through the relation
Formulas (2) and (4) are well known and follow easily by induction. Jackson [3] introduced an integral denoted by
as a right inverse of the q-derivative. It is defined by
where
provided that the series at the right-hand side of (6) converges at and b. In [4], Hahn defined the q-integration for a function f over and , , by
respectively, provided that the series converges absolutely. Al-Salam [5] defined a fractional q-integral operator by
where , as a generalization of the q-Cauchy formula
which he introduced in [6] for a positive integer n. Using (7), we can write (8) explicitly as
or in a more simple form
Using (2), we can prove
This paper is organized as follows. In Section 2, we mention in brief some known fractional and q-fractional Leibniz formulae. In Section 3, we generalize the fractional q-Leibniz formula of the Riemann-Liouville fractional q-derivative introduced by Agarwal in [7]. Finally, in Section 4, we extend the fractional q-Leibniz formula introduced by Purohit [8] for the q-Weyl derivatives of nonnegative integral orders to any real order.
2 Fractional and q-fractional Leibniz formulas
The Riemann-Liouville fractional q-integral operator is introduced by Al-Salam in [5] and later by Agarwal in [9] and defined by
Using (6), (12) reduces to
which is valid for all α. The Riemann-Liouville fractional q-derivative of order α, , is defined by
For the definition of Caputo fractional q-derivatives, see [10]. See also [11] for more applications. Liouville [12] introduced the fractional Leibniz rule
where
is the familiar Riemann-Liouville integral operator. While Liouville used Fourier expansions in obtaining (14), Grünwald [13] and Letnikov [14] obtained (14) by a different technique. Other extensions and proofs are in the work of Watanabe [15], Post [16], Bassam [17], and Gaer-Rubel [18]. In a series of papers [19–23], Osler introduced several generalizations of (14). For example, in [19] Osler introduced the fractional Leibniz formula
which coincides with (14) when we set , and replace α with −α in (15). For an extensive study of the fractional calculus and its applications in physics and control theory, see [24–28]. There are two q-analogues of the fractional Leibniz rule (14). Al-Salam and Verma [29] introduced the fractional Leibniz formula
formally. An analytic proof of (16) is introduced in [10] where the following theorem is introduced.
Theorem 2.1 Let be an entire function with q-exponential growth of order k, , and a finite type δ, . Let V be a function that satisfies
Then (16) holds for and .
For the definition of the q-exponential growth, see [30]. In [7], Agarwal introduced the following fractional q-Leibniz formula.
Theorem 2.2 Let U and V be two analytic functions which have power series representations at with radii of convergence and , respectively, and . Then
Proof See [7]. □
Recently, Purohit [31] used (17) to derive a number of summation formulae for the generalized basic hypergeometric functions. In the following section, we introduce a generalization of Agarwal’s fractional q-Leibniz formula (17). Let and . In the following, we say that a function if
In [8], Purohit derived a q-extension of the Leibniz rule for q-derivative via the Weyl q-derivative operator defined in (8). He proved that for a nonnegative integer α,
where , , u and v are analytic functions having a power series expansion at with radii of convergence ρ, , and . Purohit established some summation formulae as an application of the fractional Leibniz formula (18) which can be represented as
where we used
3 A generalization for Agarwal’s fractional q-Leibniz formula
In this section we introduce a q-analogue of the fractional Leibniz formula (15) when . The case of the derived fractional q-Leibniz formula is the fractional q-Leibniz formula (17) introduced by Agarwal [7].
Theorem 3.1 Let G be a branch domain of the logarithmic function. Let a, b be complex numbers and R be a positive number. Let u and v be analytic functions in the disk . Let U and V be defined in through the relations
If and are in , then
where , and .
Remark 3.2 It is worthwhile to notice that if we set in (21), we obtain Agarwal’s fractional Leibniz rule (17) with less restrictive conditions on the functions and . Actually, the special case of Theorem 3.1 is an extension of the result given by Manocha and Sharma in [32].
Proof Since V, UV are in , then
From (13) we obtain
Substituting with
into (22), we obtain
The existence of UV in the space guarantees that the series in (22) or in (23) converges absolutely for all . Replace x in (4) with ξ and then let
Consequently,
Then substituting (24) into (23), we get
Using (2), we obtain
Therefore, since is analytic in , there exists such that
Consequently,
Set . Then substituting (27) into (25), we obtain
The last series converges for all since . Consequently, the series in (28) is absolutely convergent, and we can interchange the order of summations in (25). This leads to
Since
and
the substitution with the last two identities in (29) gives
and the theorem follows. □
Example 3.3 Let γ, λ, μ, and α be complex numbers satisfying
Then
Proof We prove the identity by using Theorem 3.1. Take and . Then
Hence,
and
Then applying Theorem 3.1 gives
On the other hand,
Equating (33) and (34) gives (30). □
Example 3.4 For complex numbers a, b, A, B, d, and D such that , , and ,
for .
Proof
The previous identity follows by taking
and applying Theorem 3.1 with
Then using (3), we obtain
In addition,
and
Substituting with (36)-(38) into (21), we obtain
On the other hand,
Combining (39) and (40), we obtain (35). □
4 A q-extension of the Leibniz rule via Weyl-type of a q-derivative operator
In this section, we prove that the q-expansion in (18) can be derived for any . The proof we introduce is completely different from the one introduced by Purohit for nonnegative integer values of α. We start with characterizing a sufficient class of functions for which exists for some α.
Definition 4.1 Let and let f be a function defined on a -geometric set A. We say that f is of class if there exists , such that
Proposition 4.2 If , then exists for any function f defined on . If and , then exists.
Proof If , then by (11), exists for any functions f defined on a . If and , then for each , there exists a constant , C depends on x and α, such that
Applying the previous inequality in (10) gives
□
In the following, we define a sufficient class of functions for which exists for all α.
Definition 4.3 Let f be a function defined on a -geometric set A. We say that f is in the class if there exist and such that for each ,
It is clear that if , then for all α. The spaces and are q-analogues of the spaces of fairly good functions and good functions, respectively, introduced by Lighthill [[33], p.15], see also [[34], Chapter VII].
Example 4.4 An example of a function in a class is any function of the form
where is a polynomial of degree n and a is a constant such that for all .
The keynotes in proving the generalization of Purohit q-fractional Leibniz formula are two identities. The first one is
which holds for any when or holds when . The proof of (41) follows from (10) by replacing α with −α, x with z, and setting . The second identity follows from the formula (4) with q replaced with and x with z. That is,
where we use [[1], Eq. (I.47)]
The identity in (41) leads to the following result.
Lemma 4.5 Let p and α be such that and . Let G be the principal branch of the logarithmic function and let . Assume that
is analytic on . Let
Then exists for all and is equal to
Proof From (10) we find that
From the assumptions of the present lemma, we can easily deduce that the double series in (44) is absolutely convergent for all . Hence, we can interchange the order of summations in (44). This and the q-binomials theorem [[1], Eq. (1.3.2)] give
Simple manipulations give (43). □
Lemma 4.6 Let p, α, G, U, , and be as in Lemma 4.5. Then
Proof The proof is easy and is omitted. □
Theorem 4.7 Let U and V be functions defined on a -geometric set A and let . Assume that and , . Then
for all and for all . If , then (45) may not hold for all α on ℝ but only for α in a subdomain of ℝ.
Proof Let be arbitrary but fixed. Since , then
From (10),
Applying (42) with yields
From the assumptions on the function U, there exists a constant and such that
Using (2) with ( instead of q), we obtain
Consequently, the double series on (47) is bounded from above by
where we applied the identity cf., e.g., [[1], p.11],
Now, it is clear that if , then the series on the most right-hand side of (48) is convergent for all . On the other hand, it is convergent only for when . Therefore, we can interchange the order of summation in the series on the right-hand side of (47). This gives
But
Combining this latter identity with (50) yields the theorem. □
Example 4.8 Let γ, λ, μ, and α be complex numbers satisfying
Then
Proof We prove the identity by using Theorem 3.1. Take and . Then
Hence,
and
Then applying Theorem 3.1 gives
On the other hand,
Equating (54) and (55) gives (51). □
Example 4.9 For complex numbers a, b, A, B, d, and D such that , , and ,
for .
Proof
The previous identity follows by taking
and applying Theorem 3.1 with
Then using (3), we obtain
In addition,
and
Substituting with (57)-(59) into (21), we obtain
On the other hand,
Combining (60) and (61), we obtain (56). □
References
Gasper G, Rahman M: Basic Hypergeometric Series. 2nd edition. Cambridge University Press, Cambridge; 2004.
Ismail MEH: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge; 2005.
Jackson FH: On q -definite integrals. Q. J. Pure Appl. Math. 1910, 41: 193-203.
Hahn W: Beiträge zur Theorie der Heineschen Reihen. Math. Nachr. 1949, 2: 340-379. (German) 10.1002/mana.19490020604
Al-Salam WA: Some fractional q -integrals and q -derivatives. Proc. Edinb. Math. Soc. 1966/1967, 2(15):135-140.
Al-Salam WA: q -analogues of Cauchy’s formulas. Proc. Am. Math. Soc. 1966, 17: 616-621.
Agarwal RP: Fractional q -derivative and q -integrals and certain hypergeometric transformations. Ganita 1976, 27(1-2):25-32.
Purohit SD: On a q -extension of the Leibniz rule via Weyl type of q -derivative operator. Kyungpook Math. J. 2010, 50(4):473-482. 10.5666/KMJ.2010.50.4.473
Agarwal RP: Certain fractional q -integrals and q -derivatives. Proc. Camb. Philos. Soc. 1969, 66: 365-370. 10.1017/S0305004100045060
Annaby MH, Mansour ZS Lecture Notes of Mathematics 2056. In q-Fractional Calculus and Equations. Springer, Berlin; 2012.
Abdeljawad T, Baleanu D: Caputo q -fractional initial value problems and a q -analogue of Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(12):4682-4688. 10.1016/j.cnsns.2011.01.026
Liouville J: Mèmoire sur le calcul des différentielles à indices quelconques. J. Éc. Polytech. 1832, 13: 71-162.
Grünwald AK: Über begrenzte derivationen und deren anwedung. Z. Angew. Math. Phys. 1867, 12: 441-480.
Letinkov AV: Theory of differentiation of fractional order. Mat. Sb. 1868, 3: 1-68.
Watanabe Y: Notes on the generalized derivative of Riemann-Liouville and its application to Leibniz’s formula. I and II. Tohoku Math. J. 1931, 34: 8-41.
Post EL: Generalized differentiation. Trans. Am. Math. Soc. 1930, 32: 723-781. 10.1090/S0002-9947-1930-1501560-X
Bassam MA: Some properties of Holmgren-Riez transform. Ann. Sc. Norm. Super. Pisa 1961, 15(3):1-24.
Gaer MC, Rubel LA: The fractional derivative and entire functions. Lecture Notes in Math. 457. In Fractional Calculus and Its Applications(Proc. Internat. Conf., Univ. New Haven, West Haven, Conn., 1974). Lecture Notes in Math. Springer, Berlin; 1975:171-206.
Osler TJ: Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM J. Appl. Math. 1970, 18(3):658-674. 10.1137/0118059
Osler TJ: Fractional derivatives and Leibniz rule. Am. Math. Mon. 1971, 78(6):645-649. 10.2307/2316573
Diaz JB, Osler TJ: Differences of fractional order. Math. Comput. 1974, 28(125):185-202.
Osler TJ: The integral analogue of the Leibniz rule. Math. Comput. 1972, 26(120):903-915.
Osler TJ: A correction to Leibniz rule for fractional derivatives. SIAM J. Math. Anal. 1973, 4: 456-459. 10.1137/0504040
Herrmann R: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore; 2011.
Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity, and Chaos 3. In Fractional Calculus Models and Numerical Methods. World Scientific, Singapore; 2012.
Baleanu D, António J, Machado T, Luo ACJ: Fractional Dynamics and Control. Springer, Berlin; 2012.
Golmankhaneh AK: Investigations in Dynamics: With Focus on Fractional Dynamics. LAP Lambert Academic Publishing, Saarbrücken; 2012.
Hilfer R (Ed): Applications of Fractional Calculus in Physics. World Scientific, Singapore; 2000.
Al-Salam WA, Verma A: A fractional Leibniz q -formula. Pac. J. Math. 1975, 60(2):1-9. 10.2140/pjm.1975.60.1
Ramis JP: About the growth of entire functions solutions of linear algebraic q -difference equations. Ann. Fac. Sci. Toulouse 1992, 1(6):53-94.
Purohit SD: Summation formulae for basic hypergeometric functions via fractional q -calculus. Matematiche 2009, 64(1):67-75.
Manocha HL, Sharma BL: Fractional derivatives and summation. J. Indian Math. Soc. 1974, 38(1-4):371-382.
Lighthill MJ: Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press, New York; 1960.
Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.
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This research is supported by NPST Program of King Saud University; project number 10-MAT1293-02.
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Mansour, Z.S. Generalizations of fractional q-Leibniz formulae and applications. Adv Differ Equ 2013, 29 (2013). https://doi.org/10.1186/1687-1847-2013-29
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DOI: https://doi.org/10.1186/1687-1847-2013-29