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 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?
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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]](Images/jd_gr_4.gif)
Any additional examples would be greatly appreciated.
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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 .
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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 ![[Graphics:Images/jd_gr_15.gif]](Images/jd_gr_15.gif)
for ?
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6) Continuation of derivatives of the psifunction by order
How is it possible to extend the definition of the derivatives of the psifunction
![[Graphics:Images/jd_gr_19.gif]](Images/jd_gr_19.gif)
from integer parameters to arbitrary complex values ? Such an extension might, for instance, be provided by the formula
![[Graphics:Images/jd_gr_22.gif]](Images/jd_gr_22.gif)
where
![[Graphics:Images/jd_gr_23.gif]](Images/jd_gr_23.gif)
How do we represent the coefficient as an analytical function of in a natural way?
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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 :
.
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4) Formulas for Mathieu functions and their characteristics
and
We are looking for closed formulas for the Mathieu functions and their characteristics
and .
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3) The inverse function for
The equation has the solution .
Is there a good way to express the solution of the equation ?
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2) Series for the inverse error function
We are searching for general formulas for the series expansion of the inverse error
function around 0.
![[Graphics:Images/jd_gr_45.gif]](Images/jd_gr_45.gif)
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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.
![[Graphics:Images/jd_gr_48.gif]](Images/jd_gr_48.gif)
Question 1: How do we express coefficients
through and in closed form?
![[Graphics:Images/jd_gr_52.gif]](Images/jd_gr_52.gif)
Question 2: How do we express coefficients through
and
in closed form? It may be helpful to take into account that
![[Graphics:Images/jd_gr_56.gif]](Images/jd_gr_56.gif)
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