Index: combfunc.spad.pamphlet
===================================================================
--- combfunc.spad.pamphlet	(revision 580)
+++ combfunc.spad.pamphlet	(working copy)
@@ -41,11 +41,6 @@
       ++ formal product;
 
 @
-The latest change allows Axiom to reduce
-\begin{verbatim}
-   sum(1/i,i=1..n)-sum(1/i,i=1..n) 
-\end{verbatim}
-to reduce to zero.
 <<package COMBF CombinatorialFunction>>=
 )abbrev package COMBF CombinatorialFunction
 ++ Provides the usual combinatorial functions
@@ -703,21 +698,33 @@
     Gamma   : F -> F
       ++ Gamma(f) returns the formal Gamma function applied to f
     Gamma   : (F,F) -> F
-      ++ Gamma(a,x) returns the incomplete Gamma function applied to a and x
+      ++ Gamma(a,x) returns the incomplete Gamma function applied to a and x.
+      ++ Concerning differentiation, it is regarded as a function in the second
+      ++ argument only.
     Beta:      (F,F) -> F
       ++ Beta(x,y) returns the beta function applied to x and y
     digamma:   F->F
       ++ digamma(x) returns the digamma function applied to x 
     polygamma: (F,F) ->F
-      ++ polygamma(x,y) returns the polygamma function applied to x and y
+      ++ polygamma(x,y) returns the polygamma function applied to x and y. 
+      ++ Concerning differentiation, it is regarded as a function in the second
+      ++ argument only.
     besselJ:   (F,F) -> F
-      ++ besselJ(x,y) returns the besselj function applied to x and y
+      ++ besselJ(x,y) returns the besselj function applied to x and y.
+      ++ Concerning differentiation, it is regarded as a function in the second
+      ++ argument only.
     besselY:   (F,F) -> F
-      ++ besselY(x,y) returns the bessely function applied to x and y
+      ++ besselY(x,y) returns the bessely function applied to x and y.
+      ++ Concerning differentiation, it is regarded as a function in the second
+      ++ argument only.
     besselI:   (F,F) -> F
-      ++ besselI(x,y) returns the besseli function applied to x and y
+      ++ besselI(x,y) returns the besseli function applied to x and y.
+      ++ Concerning differentiation, it is regarded as a function in the second
+      ++ argument only.
     besselK:   (F,F) -> F
-      ++ besselK(x,y) returns the besselk function applied to x and y
+      ++ besselK(x,y) returns the besselk function applied to x and y.
+      ++ Concerning differentiation, it is regarded as a function in the second
+      ++ argument only.
     airyAi:    F -> F
       ++ airyAi(x) returns the airyai function applied to x 
     airyBi:    F -> F
@@ -837,7 +844,16 @@
         ahalf * (besselI (n-1,x) + besselI (n+1,x))
     iBesselKGrad(l: List F): F ==
         n := first l; x := second l
-        ahalf * (besselK (n-1,x) + besselK (n+1,x))
+        - ahalf * (besselK (n-1,x) + besselK (n+1,x))
+
+@
+The formulas above for the Bessel functions can be found in Milton Abramowitz
+and Irene A. Stegun, eds.  (1965). Handbook of Mathematical Functions with
+Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4,
+Equations~9.1.27 and 9.6.26.  Up to [[patch--50]] the formula for $K$ missed
+the minus sign.  (Issue~\#355)
+
+<<package FSPECF FunctionalSpecialFunction>>=
     ipolygammaGrad(l: List F): F ==
         n := first l; x := second l
         polygamma(n+1, x)