10) Historical information about mathematical functions
We are looking for more complete historical information about mathematical functions.
For each , who introduced
or its notation, and when? Who made essential progress in its developmentby finding basic
formulas, extending its domain of definition, defining key conventions, and so on....
A particular example: who came up with the notation for the double
factorial, and when?
9) More complete tables of particular cases of generalized
hypergeometric functions
We are creating the most complete tables possible of particular cases of the generalized
hypergeometric functions
Any additional examples would be greatly appreciated.
8) Differential equations for the theta functions
We are looking for ordinary, nonlinear differential equations for the elliptic
theta functions with respect to for fixed .
7) Generalizations of Euler and Bernoulli polynomials,
Stirling numbers,
and partitions numbers for noninteger argument
How is it possible to define the values of Euler and Bernoulli polynomials
and , Stirling numbers
and , and partitions
and
for ?
6) Continuation of derivatives of the psifunction by order
How is it possible to extend the definition of the derivatives of the psifunction
from integer parameters to arbitrary complex values ? Such an extension might, for instance, be provided by the formula
where
How do we represent the coefficient as an analytical function of in a natural way?
5) Values of derivatives of the zeta function
We are looking for closedform expressions for the derivatives of the zeta function
of arbitrary order
for integer values of argument :
.
4) Formulas for Mathieu functions and their characteristics
and
We are looking for closed formulas for the Mathieu functions and their characteristics
and .
3) The inverse function for
The equation has the solution .
Is there a good way to express the solution of the equation ?
2) Series for the inverse error function
We are searching for general formulas for the series expansion of the inverse error
function around 0.
1) Series for the gamma function
We are searching for general formulas for the series expansion of the gamma function
near its regular points and poles .
These formulas should include the psifunctions and their derivatives.
Question 1: How do we express coefficients
through and in closed form?
Question 2: How do we express coefficients through
and
in closed form? It may be helpful to take into account that
