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10) Historical information about mathematical functions

We are looking for more complete historical information about mathematical functions. For each [Graphics:Images/jd_gr_1.gif], who introduced [Graphics:Images/jd_gr_2.gif] 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 [Graphics:Images/jd_gr_3.gif] 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 [Graphics:Images/jd_gr_4.gif] Any additional examples would be greatly appreciated.





8) Differential equations for the theta functions [Graphics:Images/jd_gr_5.gif]

We are looking for ordinary, nonlinear differential equations for the elliptic theta functions [Graphics:Images/jd_gr_6.gif] with respect to [Graphics:Images/jd_gr_7.gif] for fixed [Graphics:Images/jd_gr_8.gif].





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 [Graphics:Images/jd_gr_10.gif] and [Graphics:Images/jd_gr_11.gif], Stirling numbers [Graphics:Images/jd_gr_12.gif] and [Graphics:Images/jd_gr_13.gif],  and partitions [Graphics:Images/jd_gr_14.gif] and [Graphics:Images/jd_gr_15.gif][Graphics:Images/jd_gr_16.gif] for [Graphics:Images/jd_gr_17.gif]?





6) Continuation of derivatives of the psi­function [Graphics:Images/jd_gr_18.gif] by order

How is it possible to extend the definition of the derivatives of the psi­function

[Graphics:Images/jd_gr_19.gif]

from integer parameters [Graphics:Images/jd_gr_20.gif] to arbitrary complex values [Graphics:Images/jd_gr_21.gif]? Such an extension might, for instance, be provided by the formula

[Graphics:Images/jd_gr_22.gif]

where

[Graphics:Images/jd_gr_23.gif]

How do we represent the coefficient [Graphics:Images/jd_gr_24.gif] as an analytical function of [Graphics:Images/jd_gr_25.gif] in a natural way?





5) Values of derivatives of the zeta function [Graphics:Images/jd_gr_27.gif]

We are looking for closed­form expressions for the derivatives of the zeta function [Graphics:Images/jd_gr_28.gif] of arbitrary order [Graphics:Images/jd_gr_29.gif] for integer values of argument [Graphics:Images/jd_gr_30.gif]:   [Graphics:Images/jd_gr_31.gif].





4) Formulas for Mathieu functions and their characteristics [Graphics:Images/jd_gr_33.gif] and [Graphics:Images/jd_gr_34.gif]

We are looking for closed formulas for the Mathieu functions and their characteristics [Graphics:Images/jd_gr_35.gif] and [Graphics:Images/jd_gr_36.gif].





3) The inverse function for [Graphics:Images/jd_gr_39.gif]

The equation [Graphics:Images/jd_gr_40.gif] has the solution [Graphics:Images/jd_gr_41.gif]. Is there a good way to express the solution of the equation [Graphics:Images/jd_gr_42.gif]?





2) Series for the inverse error function

We are searching for general formulas for the series expansion of the inverse error function [Graphics:Images/jd_gr_44.gif] around 0.

[Graphics:Images/jd_gr_45.gif]




1) Series for the gamma function

We are searching for general formulas for the series expansion of the gamma function [Graphics:Images/jd_gr_46.gif] near its regular points and poles [Graphics:Images/jd_gr_47.gif]. These formulas should include the psi­functions and their derivatives.

[Graphics:Images/jd_gr_48.gif]

Question 1: How do we express coefficients [Graphics:Images/jd_gr_49.gif] through [Graphics:Images/jd_gr_50.gif] and [Graphics:Images/jd_gr_51.gif] in closed form?

[Graphics:Images/jd_gr_52.gif]

Question 2: How do we express coefficients [Graphics:Images/jd_gr_53.gif] through [Graphics:Images/jd_gr_54.gif] and [Graphics:Images/jd_gr_55.gif] in closed form?  It may be helpful to take into account that

[Graphics:Images/jd_gr_56.gif]


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