BERNOULLI ’ S n-FORMULA AND n MULTI-SERIES BY THE GENERALIZED DIFFERENCE EQUATION

The Fractional Calculus is currently a very important research field in several different areas: physics (including classical and quantum mechanics and thermodynamics), chemistry, biology, economics and control theory ([5], [6], [7]). In 1989, K.S.Miller and Ross [8] introduced the discrete analogue of the Riemann-Liouville fractional derivative and proved some properties of the fractional difference operator. The main definition of fractional difference equation (as done in [8]) is the ν fractional sum of f(t) by


Introduction
The Fractional Calculus is currently a very important research field in several different areas: physics (including classical and quantum mechanics and thermodynamics), chemistry, biology, economics and control theory ( [5], [6], [7]).In 1989, K.S.Miller and Ross [8] introduced the discrete analogue of the Riemann-Liouville fractional derivative and proved some properties of the fractional difference operator.The main definition of fractional difference equation (as done in [8]) is the ν fractional sum of f (t) by where ν > 0. On the other hand, when ν = m is a positive integer, if we replace Received: December 23, 2013 c 2014 Academic Publications, Ltd.
url: www.acadpubl.eu§ Correspondence author the function f (t) by u(k) and ∆ by ∆ ℓ defined as ∆ ℓ u(k) = u(k + ℓ) − u(k), (1) becomes Let ℓ i > 0, u(k) be real valued function on [0, ∞), u(k) = 0 for all k ∈ (−∞, 0], [k/ℓ i ] be the integer part of k/ℓ i , where u and in general u Substituting u , u which is a numerical solution of the generalized difference equation By denoting R.H.S of (4) as In particular, when n = 1, u(k) When (k) given in (2).We find that, by expanding the terms, u (k) is independent of the order of the parameters ℓ 1 ,ℓ 2 ,• • • ,ℓ n .There are direct formulas to find the n series when There is no direct formula to find the sum of n multi-series in the existing literature.We find that the n multi-series ℓ [1,n] u( k) is the numerical solution as well as the complete solution (closed form solution with lower limits) of equation ( 5), so we call u ℓ [1,n] (k) as the complete solution and ℓ [1,n] u( k) as the numerical solution of (5).Hence in this paper, we obtain the numerical-complete relation (6) and derive n multi-series with Bernoulli's n-formula.

Main Results And Applications
Here, by introducing Striling numbers of third kind and expressing the polynomial factorial k Proof.The proof follows from (11) and first, second terms of (8).
a set of positive reals and and k (1) ℓ 2 from second relation of (9), we both sides of the above and applying (12), we obtain ∆ −1 . Now the proof is completed by taking ∆ −1 ℓ i and applying second relation of ( 9) and ( 12) for i = 4, 5, • • • , n respectively.
Proof.Taking ν = p o and ℓ = ℓ 1 in second relation of (9), we get The proof follows by repeatedly applying (12), ∆ −1 ℓ i and second relation of ( 9) Proof.Replacing n by p o , r by p 1 and ℓ by ℓ 1 in second relation of ( 8), we Applying ∆ −1 ℓ 1 and using second relation of (9), we get Using ( 12) in ( 16), we obtain The proof of (15) follows by contuining this process n times, The following Theorem is the generalization of Theorem 2.2.
Proof.From first relation of ( 9), we have .
Taking ∆ t 2 ℓ 2 on both sides of the above equation, we obtain ).
Applying (12), we get The proof follows by repeating this process n times.
on both sides of the above equation, we get The proof follows by repeatedly taking ∆ Theorem 3.10.Bernoulli's n Formula to the product k ) Proof.Since (k ) where (k (a The following theorem gives complete solution of equation ( 5).
Theorem 3.11.Consider the functions u given in the notations and above.Assume that for each i, be any closed form solution of the difference equation ∆ is the complete solution of equation (5).
Proof.Since 1 = k (0) , applying the limit from , which is the complete solution of equation ( 5) for n = 1.Taking ∆ −1 ℓ 2 on both sides and applying the limits from ℓ 2 (k) to k and keeping ∆ −1 ℓ 1 u(ℓ 1 (k)) as a constant, we obtain which is the complete solution of the equation ( 5), and it can be expressed as u In the R.H.S of the above expression, second term is associated to {m 1 } = {1} ∈ 1(J 2 ), third term to {m 1 } = {2} ∈ 1(J 2 ) and the fourth term to {m 1 , m 2 } = {1, 2} ∈ 2(J 2 ).Taking ∆ −1 ℓ 3 on u 2 (k), applying the limits ℓ 3 (k) and k, and as u(ℓ 2 (k)) are constants, we get a relation of the form u (ℓ 3 (k)), and this relation can be expressed as which is a complete solution of the equation ( 5) for n = 3.As all the lower limit values are constants, the proof is completed by taking ∆ −1 ℓ i and applying the limit from ℓ i (k) to k on u The following theorem gives the numerical solution of the equation (5).
Theorem 3.12.Consider the terms of Theorem 3.11.Then, is the numerical solution of the difference equation ( 5).

Lemma 2 . 1 .
[2] If s n r and S n r are the Stirling numbers of the first and second kinds respectively, k ℓ b , r = 1, 2, • • • n and Stirling numbers of third kind, we derive the Bernoulli's n-formula and n multi-series to product of arithmetic and geometric functions with examples.Definition 3.1.Let 1 ≤ p ≤ n, The Stirling number of third kind for the pair of positive reals ℓ a and ℓ b is defined by

Lemma 3 . 8 .
Let t ′ i s and p o are positive integers.Then, n i=1