On a general class of trigonometric functions and Fourier series†

†Dedicated to Prof. J. Bellandi Filho, our teacher and advisor, on his 65th birthday.

Abstract

We discuss a general class of trigonometric functions whose corresponding Fourier series can be used to calculate several interesting numerical series. Particular cases are presented.

†Dedicated to Prof. J. Bellandi Filho, our teacher and advisor, on his 65th birthday.

Keywords:

• Fourier series
• sums
• numerical series

Acknowledgements

We are grateful to Prof. J. Vaz Jr and to Dr J. Emílio Maiorino for several and useful discussions. ECO is also grateful to Fapesp (06/52475-8) for a research grants.

Notes

†Dedicated to Prof. J. Bellandi Filho, our teacher and advisor, on his 65th birthday.

Notes

1. We also note that another possible class of functions is g(x) = sin^ℓ x with ℓ = 1, 2, 3, … expanded on the interval a ≤ x ≤ b where a and b must be chosen in a convenient way. Here, we do not discuss this case.

2. In the calculation of the Fourier coefficient a _m , we have two possibilities. One of them produces the value m = k − n; as n > k and m is a positive integer, this case must be omitted. The other one produces m = n − k and then we have a _m ≡ a _n−k .

3. Note that we do not have an infinite series. The unique term contributing to the sum is m = n − k.

4. Note that the equation for a _m gives the same a ₀ for m = 0.