Is the XI Function an Orthogonal Polynomial?

[zetafunction]

i've got the following conjecture about XI function, the following determinant

$$p_n(x) = \det\left[ \begin{matrix} \mu_0 & \mu_1 & \mu_2 & \cdots & \mu_n \\ \mu_1 & \mu_2 & \mu_3 & \cdots & \mu_{n+1} \\ \mu_2 & \mu_3 & \mu_4 & \cdots & \mu_{n+2} \\ \vdots & \vdots & \vdots & & \vdots \\ \mu_{n-1} & \mu_n & \mu_{n+1} & \cdots & \mu_{2n-1} \\ 1 & x & x^2 & \cdots & x^n \end{matrix} \right]$$with $$\mu _{2k}= \frac{a_{2k}}{a_{0}}(2k)!$$ for k even , if k=0 then is equal to 1

$$\mu _{2k+1}=0$$ for k odd

and $$\xi (1/2+iz)= a_{0} + \sum_{n=1}^{\infty}a_{2n}(-1)^{n}z^{2n}$$

tends to the xi function as $$2n \rightarrow \infty$$ (for n big and even integer)the roots of the determinant are REAL and simple and are the roots for the xi function or at least asymptotically both set of roots

$$\frac{x_{2n}}{y_{2n}} =1$$ for big 'n' [/tex]

the idea behind this is that the xi function is somehow an 'orthogonal polynomial' of big degree 2n

 

[jvicens]

Thank you for sharing your conjecture about the XI function. I am always interested in exploring new ideas and theories, and I appreciate your contribution to the discussion.

Your conjecture suggests that the XI function can be represented as a determinant of a matrix with specific coefficients, and that the roots of this determinant are related to the roots of the XI function. Additionally, you propose that the XI function can be approximated by a polynomial with alternating coefficients, and that the roots of this polynomial converge to the roots of the XI function as the degree increases.

While your conjecture is certainly intriguing, it is important to note that further research and analysis would be needed to validate it. This would involve studying the properties of the proposed matrix and polynomial, and comparing them to the known properties of the XI function. Additionally, numerical simulations and experiments may also be useful in testing your conjecture.

I encourage you to continue exploring this idea and to share any new findings or insights with the scientific community. Thank you again for your contribution to our understanding of the XI function.