eu FINITE AND INFINITE MULTI-SERIES TYPE SOLUTIONS OF GENERALIZED MIXED DIFFERENCE EQUATION

In this paper, we obtain the solutions of a generalizedi-mixed difference equa- tion in closed, finite and infinite multi-series forms. By equating closed form with multi-series type solutions of thei-mixed difference equation, we obtain the values of certain types of infinite and finite multi-series. Suitable examples are provided to illustrate the main results.

An equation involving both ∆ and ∆ α is called a mixed difference equation.Oscillatory behaviour of solutions for a few mixed difference equations were discussed in [3,4,6,12].An equation involving ∆ ℓ and ∆ α(ℓ) is called as the generalized mixed difference equation.
The higher order generalized α i -difference equation becomes a generalized mixed difference equation if α i = 1 for some i and n ≥ 2.
The equation ( 1) has three types of solutions which are closed, finite and infinite multi-series forms.

Preliminaries
In this section, we present some basic definitions, results and examples on α imixed difference equation.Throughout this paper, we assume that k ∈ [0, ∞), ℓ i > 0, α i = 0 and we denote ∆ Here . The solution in infinite series form of the first order difference equation is given in Example 2.6.
Lemma 2.4.[11] The first order generalized α-difference equation has a solution in the finite summation form as where has a solution in the infinite series form as Proof.Taking α = α i and ℓ = ℓ i in (4), we get Replacing k by k + l i in (8), we obtain From ( 9), ∆ −1 (8), we arrive (7).
The following example illustrates the existence of the solution in the infinite series form.
Example 2.6.If ℓ i > 0, α i ≥ 1, a > 1 and s < 0, then we have Proof.The proof follows by equating the infinite series form solution given in Lemma 2.5 and the closed form solution in Example 2.2.

Main Results
In this section, we obtain the values of certain types of finite and infinite alpha multi-series by equating the closed form with finite and infinite multi-series solutions of the generalized α i -difference equation (1).Suitable examples are presented for second order mixed difference equation.
is called an infinite multi-series solution of the equation ( 1) and the summation of the form is called a finite multi-series solution of the equation ( 1).
If there exists a function v(k), other than the finite and infinite multiseries solutions, such that ∆ α Theorem 3.3.(Infinite alpha multi-series) If for t = 1, 2, ..., n, lim is an infinite multi-series solution of the equation (1).
which is an infinite multi-series solution of the equation ( 1) for each α i = 1.
Proof.The proof follows by taking each α i = 1 in (11).
The following example gives the value of an infinite series.
The following example illustrates the existence of infinite and finite multiseries solutions of a second order mixed difference equation.
From (26), the equation ( 33) has an infinite multi-series solution Now, we discuss ∆ −1 we get Similarly, since the mixed difference equation (33) for u(k) = k has a closed form solution Taking u(k) = k in (34), we get which is not convergent.Hence, (33) does not have an infinite multi-series solution.
Next we consider finite multi-series solution of (33) when u(k) = k.
Similarly one can obtain closed, infinite and finite multi-series solutions of the difference equation ( 1) for various functions u(k).

Conclusion
In general, closed form solution of the equation ( 1) is not possible always.When closed form solution is not possible we can go for either infinite summation form solution or finite summation form solution according to our need.If infinite series solution is not convergent, then we can find the finite series solution always.Theorem 3.11 gives an infinite multi-series solution and the Theorem (3.12) shows that the equation (1) has a finite multi-series solution for any function u(k).

Definition 3 . 1 .
The solution of the form