            /**  Digamma and Polygamma (Psi) functions **/
 
/*A: S.Wolfram */
/*S: California Institute of Technology */
/*D: July 1981 */

SPsi _: Ldist
 
SPsi[1]:   Psi[$z] -> Psi[$z+1] -1/$z
                                                /*R: MOS p. 14 */
 
SPsi[2]:   Psi[$z] -> Psi[1-$z]-Pi Cot[Pi $z]
                                                /*R: MOS p. 14 */
 
SPsi[3]:   Psi[$z] -> Psi[-$z]-1/$z-Pi Cot[Pi $z]
                                                /*R: MOS p. 14 */
 
SPsi[4]:   Psi[$z] -> Psi[-$z+2]+1/($z-1)-Pi Cot[Pi($z-1)]
                                                /*R: MOS p. 14 */
 
SPsi[5]:   Psi[$z] -> Psi[-$z+1]+Pi Tan[Pi($z-1/2)]
                                                /*R: MOS p. 14 */
 
SPsi[6] :  Psi[$z] -> Psi[$z-1] + 1/($z-1)
 
SPsi[7]:   Psi[$n_=Natp[$n],1] -> (-1)^($n+1)$n!*Zeta[$n+1]
                                                /*R: AS 6.4.2 */
 
SPsi[8]:   Psi[$m_=Natp[$m],$n] -> (-1)^$m $m!*(-Zeta[$m+1]+\
                Sum[#%k^(-$m-1),{#%k,1,$n-1}])
                                                /*R: AS 6.4.3 */
 
SPsi[9]:   Psi[$n_=Natp[$n],1/2] -> (-1)^($n+1) $n!*(2^($n+1)-1)Zeta[$n+1]
                                                /*R: AS 6.4.4 */
 
SPsi[10]:  Psi[$n,$z] -> Psi[$n,$z-1]+(-1)^$n $n!*($z-1)^(-$n-1)
                                                /*R: AS 6.4.6 */
 
SPsi[11]:  Psi[$n,$z] --> (-1)^($n+1)Psi[$n,1-$z]-Pi D[Cot[Pi $q],{$q,$n,$z}]
                                                /*R: AS 6.4.7 */
 
SPsi[12]:  Psi[$n,($m_=Natp[$m]) $z] -> ($n=0)Log[$m]+$m^(-$n-1)\
                Sum[Psi[$n,$z+#%k/$m],{#%k,0,$m-1}]
                                                /*R: AS 6.4.8 */
 

If[~P[_XLoadonce[Loaded]],<XLoadonce]
Loadonce[XPsiV]

_XPsi[Loaded] : 1

