Brief Note - Introduction a generalization form of n-partial sum of Hurwitz Zeta Function.
The n-th partial sums of the expressions (I) and (II) below are found in closed-form for each positive integer n and real x such that x ≠ -k and x ≠ -(k+1).
.
.
Special values
- When n tends to infinity, (I) is reduced to a simple form
.
Let x = 0,
.
- Similarly, (II) gives
. (Correct - 11/17/2009)
At x = 1,
.
Example
- It is easy to verify that for n = 2, both sides of (I) are equal for all x ≠ - 1, -2, and -3, namely
.
A Generalization Form of n-th Partial Sum of Hurwitz Zeta Function
Recall that Hurwitz Zeta Function [1] is defined for complex arguments s and a by
,
where Re(s) > 1 and Re(a) > 0.
We now define a new generalization form of n-th partial sum of Hurwitz zeta function above as follows:
.
Based on this new define, (I) and (II) are then rewritten in terms of n-th partial sum of Hurwitz zeta function for Re[a] = x, R[s] = 2, and each positive integer n as shown in the following:
(III) .
(IV) .
The identities (III) and (IV) are also true for complex number a by replacing x = a.
(November 07, 2009)
(Update formula syntax and definition - December 05, 2009)
Other related series
50 Identities of Power Summation
References
[1] http://en.wikipedia.org/wiki/Hurwitz_zeta_function
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