/** Series of rational functions  **/

/*K: Psi; digamma functions; ScriptF functions */
/*A: J Gottschalk */
/*S: University of Western Australia */
/*D: September 1984 */

/* Contains assignments for simplifying ScriptF functions . */

/*: ScriptF[$expr]
	currently describes three sums,
	ScriptF[$n]      = Sum[(-1)^%#r/(2%#r+$n),{%#r,0,Inf}]
	ScriptF[$m,$n]   = Sum[(-1)^(%#r+%#s)/((2%#r+$m)(2%#r+2%#s+$n)),\
						{{%#r,0,Inf},{%#s,0,Inf}}]
	ScriptF[{$m,$n}] = Sum[(-1)^(%#r+%#s)/((2%#r+$m)(2%#s+$n)),\
						{{%#r,0,Inf},{%#s,0,Inf}]]
    */
ScriptF_:Tier

ScriptF[$x_=If[$x=0,Prh["ScriptF[0] generated"]]] :: ScriptF[0]
ScriptF[$x_=Natp[$x-2] & Numbp[$x]] :: (ScriptF[$x]:1/($x-2)-ScriptF[$x-2])
ScriptF[$x_=    $x<0   & Numbp[$x]] :: (ScriptF[$x]:1/$x    -ScriptF[$x+2])
ScriptF[1] : Pi/4
ScriptF[2] : Log[2]/2

ScriptF[$x_=If[$x=0,Prh["ScriptF[0,$y] generated"]],$y] :: ScriptF[$x,$y]
ScriptF[$x,$y_=If[$y=0,Prh["ScriptF[$x,0] generated"]]] :: ScriptF[$x,$y]
ScriptF[$x_=Natp[$x-2] & Numbp[$x],$y_=$y=1|$y=2] :: \
   (ScriptF[$x,$y]: ScriptF[$x-2,$y]+ScriptF[{$x,$y}]-ScriptF[$y]/($x-2))
ScriptF[$x_=$x=1|$x=2,$y_=Natp[$y-2] & Numbp[$y]] :: \
   (ScriptF[$x,$y]:-ScriptF[$x,$y-2]+ScriptF[{$x,$y-2}])
ScriptF[$x_=Numbp[$x],$y_=(Natp[-$x] | Natp[-$y]) & Numbp[$y]] :: \
   (ScriptF[$x,$y]:-ScriptF[$x+2,$y+2]+ScriptF[$y]/$x)
ScriptF[$x_=Natp[$x-2] & Numbp[$x],$y_=Natp[$y-2] & Numbp[$y]] :: \
   (ScriptF[$x,$y]:-ScriptF[$x-2,$y-2]+ScriptF[$y-2]/($x-2))
ScriptF[1,1] : Catalan/2+Pi Log[2]/8
ScriptF[1,2] : Pi^2/32
ScriptF[2,1] : Pi/4-Log[2]/2+Catalan/2-Pi Log[2]/4
ScriptF[2,2] : Pi^2/48-Log[2]^2/8

ScriptF[{$x,$x_=If[$x=0,Prh["ScriptF[{0,0}] generated"]]}] :: ScriptF[{$x,$x}]
ScriptF[{$x,$y_=$x~=$y & Numbp[$x-$y]}]  :  (ScriptF[$y]-ScriptF[$x])/($x-$y)
ScriptF[{$x,$x_=Natp[-$x]  & Numbp[$x]}] :: (ScriptF[{$x,$x}]:\
   -ScriptF[{$x+2,$x+2}]+1/$x^2)
ScriptF[{$x,$x_=Natp[$x-2] & Numbp[$x]}] :: (ScriptF[{$x,$x}]:\
   -ScriptF[{$x-2,$x-2}]+1/($x-2)^2)
ScriptF[{1,1}] : Catalan
ScriptF[{2,2}] : Pi^2/48

/*R: Erdelyi Vol.1, John Gottschalk's Phd notes */

_XSeries[Loaded] : 1

