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 [, p.15], see also [, 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 [, 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
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 [, 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
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., [, 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
Combining this latter identity with (50) yields the theorem. □
Example 4.8 Let γ, λ, μ, and α be complex numbers satisfying
Proof We prove the identity by using Theorem 3.1. Take and . Then
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 ,
The previous identity follows by taking
and applying Theorem 3.1 with
Then using (3), we obtain
Substituting with (57)-(59) into (21), we obtain
On the other hand,
Combining (60) and (61), we obtain (56). □