Integrating a Continued Fraction for ζ ( n )

I show that odd (and other) zeta constants such as Apery’s constant can be obtained from integrating a continued fraction function. MAIN We start with the identity for the Mellin transform of csch(x), \int_0^\infty x^{s-1}(x)\;dx = 2^{1-s}(2^s-1)\Gamma(s)\zeta(s),\;\;(s)>1 allowing use to get a formula for ζ(n) with n > 1 an integer, for example n = 3 will give Apery’s constant. \zeta(3) = {7} \int_0^\infty x^2 (x) \; dx the next step is to convert the csch(x) function into a continued fraction, from the reference below we may write (x) = {x} - {T_0(x)}\\ T_n(x) = S_n(ix) \\ S_n(x) = (2n+2)(2n+3) - x^2 + {S_{n+1}(x)} where i is the imaginary unit. Then we may write \zeta(3) = {7} \int_0^\infty x - {6+x^2-{20+x^2-{42+x^2-{\cdots}}}}\;dx where the sequence of numbers 6, 20, 42, 72, 110, 156, 210, ⋯ is A068377, and have the general form (2n − 2)(2n − 1) here starting from n = 2. ALL POWERS We can of course repeat this for all of the odd (and even) values of ζ(n) if we wish. Does this show that these kind of constants are all irrational as they have an infinite continued fraction? Because it is being integrated over the answer is probably no. \zeta(5) = {93} \int_0^\infty x^3 - {6+x^2-{20+x^2-{42+x^2-{\cdots}}}}\;dx \zeta(7) = {5715} \int_0^\infty x^5 - {6+x^2-{20+x^2-{42+x^2-{\cdots}}}}\;dx with the general case \zeta(n) = {2^{1-n}(2^n-1)\Gamma(n)}\int_0^\infty x^{n-2} - {6+x^2-{20+x^2-{42+x^2-{\cdots}}}}\;dx N.B. If you decide to truncate the infinite continued fraction, a small divergence about x = d appears which shrinks are more terms are used. If one wanted to numerically integrate to within some accuracy, the integrals from 0 to d − ε and from d + ε to ∞ can be computed. If we do take the infinite case, we can see that ζ(n), is purely given by the integral of the closing difference between two functions, we can truncate the integral \zeta(n) = {2^{1-n}(2^n-1)\Gamma(n)}\int_0^y x^{n-2} - {6+x^2-{20+x^2-{42+x^2-{\cdots}}}}\;dx and separate \zeta(n) = {2^{1-n}(2^n-1)\Gamma(n)}\left( }{n-1} -\int_0^y {6+x^2-{20+x^2-{42+x^2-{\cdots}}}}\;dx \right) REFERENCES Continued Fraction Expansion of csch(x) (in Russian) РАЗЛОЖЕНИЕ В ЦЕПНУЮ ДРОБЬ ФУНКЦИЙ cos(x), sec(x), ch(x), sch(x), sin(x), cosec(x), sh(x), csch(x), sec2(x), cosec2(x), sch2(x), csch2 (x) С.Н. Гладковский https://ia601406.us.archive.org/10/items/ContinuedFractionExpansionForFunctionsCosxSecxChxSchx/CFMZ_sin_x_PDFreview02.pdf Mellin Transform Identity From Table of Series and Integrals ()