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 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.
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
2.2 Formulas for Beta at even number
2.3 Formulas for Beta at odd number
2.4 Formulas for Beta at complex number
is a complex number such that