Summary

1 Dirichlet Beta Generating Functions

sech x , sec x and csc x can be expanded to Fourier series and Taylor series. And if the termwise higher

order integration of these is carried out, Dirichlet Beta at a natural number are obtained.

Where, these are automorphisms which are expressed by lower betas. However, in this chapter, we stop

those so far. The work that obtain the non-automorphism formulas by removing lower betas from these is done

in the next chapter 2 Formulas for Dirichlet Beta .

In this chapter, we obtain the following polynomials from the beta generating functions of each family of sech,

sec and csc .

Where, Dirichlet Beta and Dirichlet Lambda are as follows.

Bernoulli numbers and Euler numbers are as follows.

Harmonic number is

Furthermore, if the termwise higher order differentiation of the Fourier series of each family of sech and sec

are carried out, the following expressions are obtained.

Where, is a kind of Eulderian Number and is defined as follows.

Here, removing the lower betas from the the automorphism formulas in the previous chapter, we obtain

the following non-automorphism formulas.

Where, Bernoulli numbers and Euler numbers are as follows.

And, gamma function and incomplete gamma function were as follows.

2.1 Formulas for Beta at natural number

For ,

Especially,

Example

2.2 Formulas for Beta at even number

For ,

Especially,

Example

2.3 Formulas for Beta at odd number

For ,

Especially,

2.4 Formulas for Beta at complex number

When is a complex number such that ,

For ,

Especially,

2012.04.13

K. Kono