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Extended Wang sum and associated products.

Robert Reynolds¹, Allan Stauffer¹

¹Department of Mathematics and Statistics, York University, Toronto, ON, Canada.

|October 17, 2022

Summary

This summary is machine-generated.

This study extends the Wang sum of Lerch

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Area of Science:

Number Theory
Analytic Number Theory
Special Functions

Background:

The Wang sum and exponential sums of Lerch's Zeta functions are foundational in analytic number theory.
Previous work established connections between these sums and trigonometric functions.

Purpose of the Study:

To generalize the Wang sum involving Lerch's Zeta functions to finite sums of the Hurwitz-Lerch Zeta function.
To derive new identities for sums and products involving trigonometric functions (cosine and tangent).
To establish recurrence relations for the Hurwitz-Lerch Zeta function.

Main Methods:

Extension of the Wang sum to finite sums of the Hurwitz-Lerch Zeta function.
Application of a general theorem involving finite sums over positive integers with complex parameters.
Derivation of new recurrence identities for the Hurwitz-Lerch Zeta function.

Main Results:

New finite sum and product formulae involving cosine and tangent functions.
Established recurrence identities for the Hurwitz-Lerch Zeta function with consecutive neighbors.
Evaluation of derived sum and product formulae.

Conclusions:

The generalization of the Wang sum provides a powerful tool for deriving trigonometric identities.
The derived recurrence relations offer new insights into the properties of the Hurwitz-Lerch Zeta function.
The study successfully connects number theory concepts with trigonometric function identities.