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 development---by 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?

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 .

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 ?

We are searching for general formulas for the series expansion of the inverse error function around 0.

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