Compute Sums Involving Special Functions 

Compute indefinite and definite sums involving GammaRegularized and BetaRegularized.

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X Sum[GammaRegularized[k, z], k]

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In[2]:=

X Sum[k k! GammaRegularized[k, 3], {k, 2, n}]

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In[3]:=

X Sum[3^(-k) BetaRegularized[2, k, 3] , {k, 1, \[Infinity]}]

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Create a gallery of elegant sums involving these functions.

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In[4]:=

X gammabetasums = Inactivate[{Sum[GammaRegularized[k, z], k], Sum[k^2 GammaRegularized[k, 5], k], Sum[2^(-k)*GammaRegularized[k, 2], {k, 1, Infinity}], Sum[ BetaRegularized[2, k, 3], k], Sum[7^(-k) BetaRegularized[2, k, 3] , {k, 1, Infinity}] }, Sum]; FormulaGallery[forms_List] := Module[{vals = ParallelMap[Activate, forms]}, Column[Table[ Style[forms[[r]] == vals[[r]], ScriptLevel -> 0], {r, Length[forms]}], Dividers -> All, Alignment -> "\[LongEqual]", Background -> LightBlue, Spacings -> 1]]; FormulaGallery[gammabetasums] // TraditionalForm

Out[4]//TraditionalForm=