Improve PolyLog

- https://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog3/03/01/03/0002/

Author: axkr

symja_android_library/matheclipse-core/src/main/java/org/matheclipse/core/builtin/SpecialFunctions.java

@@ -1979,6 +1979,20 @@ private IExpr polyLogSymbolic(IExpr n, IExpr p, IExpr z) {
       if (z.isZero()) {
         return F.C0;
       }
+      if (n.isOne()) {
+        int pInt = p.toIntDefault();
+        if (pInt > 1) {
+          // https://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog3/03/01/03/0002/
+          IExpr v4 = F.CN1.pow(pInt);
+          IExpr sum = F.sum(k -> F.Times(F.Power(F.Factorial(F.Plus(F.CN1, F.Negate(k), p)), F.CN1),
+              F.Power(F.CN1, k), F.Power(F.Log(F.Subtract(F.C1, z)), F.Plus(F.CN1, F.Negate(k), p)),
+              F.PolyLog(F.Plus(k, F.C2), F.Subtract(F.C1, z))), 0, pInt - 1);
+          return F.Plus(
+              F.Times(v4, F.Power(F.Factorial(p), F.CN1),
+                  F.Power(F.Log(F.Subtract(F.C1, z)), p), F.Log(z)),
+              F.Times(v4, sum), F.Zeta(F.Plus(p, F.C1)));
+        }
+      }
       if (p.isOne()) {
         return F.PolyLog(n.inc(), z);
       }

symja_android_library/matheclipse-core/src/test/java/org/matheclipse/core/system/LowercaseTestCase.java

@@ -18372,6 +18372,12 @@ public void testPolyGamma() {
   public void testPolyLog() {
     // check("PolyLog(10007,-1.5707963267948966)", //
     // "");
+    check("PolyLog(1, 3, z)", //
+        "Pi^4/90-1/6*Log(1-z)^3*Log(z)-1/2*Log(1-z)^2*PolyLog(2,1-z)+Log(1-z)*PolyLog(3,1-z)-PolyLog(\n" //
+            + "4,1-z)");
+    check("PolyLog(1, 4, z)", //
+        "1/24*Log(1-z)^4*Log(z)+1/6*Log(1-z)^3*PolyLog(2,1-z)-1/2*Log(1-z)^2*PolyLog(3,1-z)+Log(\n" //
+            + "1-z)*PolyLog(4,1-z)-PolyLog(5,1-z)+Zeta(5)");
     check("PolyLog(-42,Infinity)", //
         "Indeterminate");
     check("PolyLog(0,Infinity)", //