Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations

and Applied Analysis 3 and define recursively a∇−nf t ∫ t a a∇−n 1f τ ∇τ 2.4 for n 2, 3, . . .. Then we have the following. Proposition 2.1 Nabla Cauchy formula . Let n ∈ Z , a, b ∈ T and let f : T → R be ∇-integrable on a, b ∩ T. If t ∈ T, a ≤ t ≤ b, then a∇−nf t ∫ t a ̂ hn−1 ( t, ρ τ ) f τ ∇τ . 2.5 Proof. This assertion can be proved by induction. If n 1, then 2.5 obviously holds. Let n ≥ 2 and assume that 2.5 holds with n replaced with n − 1, that is, a∇−n 1f t ∫ t a ̂ hn−2 ( t, ρ τ ) f τ ∇τ. 2.6 By the definition, the left-hand side of 2.5 is an antiderivative of a∇−n 1f t . We show that the right-hand side of 2.5 is an antiderivative of ∫ t a ̂ hn−2 t, ρ τ f τ ∇τ . Indeed, it holds ∇ ∫ t a ̂ hn−1 ( t, ρ τ ) f τ ∇τ ∫ t a ∇̂ hn−1 ( t, ρ τ ) f τ ∇τ ∫ t a ̂ hn−2 ( t, ρ τ ) f τ ∇τ, 2.7 where we have employed the property ∇ ∫ t a g t, τ ∇τ ∫ t a ∇g t, τ ∇τ g(ρ t , t) 2.8 see 9, page 139 . Consequently, the relation 2.5 holds up to a possible additive constant. Substituting t a, we can find this additive constant zero. The formula 2.5 is a corner stone in the introduction of the nabla fractional integral a∇−αf t for positive reals α. However, it requires a reasonable and natural extension of a discrete system of monomials ̂ hn, n ∈ N0 to a continuous system ̂ hα, α ∈ R . This matter is closely related to a problem of an explicit form of ̂ hn. Of course, it holds ̂ h1 t, s t − s for all t, s ∈ T. However, the calculation of ̂ hn for n > 1 is a difficult task which seems to be answerable only in some particular cases. It is well known that for T R, it holds ̂ hn t, s t − s n n! , 2.9 4 Abstract and Applied Analysis while for discrete time scales T Z and T q {qk, k ∈ Z} ∪ {0}, q > 1, we have ̂ hn t, s ∏n−1 j 0 ( t − s j) n! , ̂ hn t, s n−1 ∏ j 0 qt − s ∑j r 0 q r , 2.10 respectively. In this connection, we recall a conventional notation used in ordinary difference calculus and q-calculus, namely, t − s n n−1 ∏ j 0 ( t − s j), t − s n q̃ t n−1 ∏