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.
math0001.png
Bernoulli numbers and Euler numbers are as follows.
math0002.png
math0003.png
Harmonic number is math0004.png
math0005.png
math0006.png
math0007.png
math0008.png
math0009.png
math0010.png
math0011.png
math0012.png
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.
math0013.png
math0014.png
math0015.png
Where, math0016.png is a kind of Eulderian Number and is defined as follows.
math0017.png
2 Formulas for Dirichlet Beta
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.
math0018.png
math0019.png
And, gamma function and incomplete gamma function were as follows.
math0020.png
2.1 Formulas for Beta at natural number
For math0021.png,
math0022.png
Especially,
math0023.png
math0024.png
Example
math0025.png
math0026.png
2.2 Formulas for Beta at even number
For math0027.png,
math0028.png
math0029.png
math0030.png
math0031.png
Especially,
math0032.png
Example
math0033.png
math0034.png
2.3 Formulas for Beta at odd number
For math0035.png,
math0036.png
math0037.png
Especially,
math0038.png
2.4 Formulas for Beta at complex number
When math0039.pngis a complex number such that math0040.png,
For math0041.png,
math0042.png
Especially,
math0043.png
2012.04.13
K. Kono
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