Generalized Bernoulli polynomials

paulmasson · Sep 10, 2023 · 931ee02 · 931ee02
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Showing 3 changed files with 54 additions and 16 deletions.
diff --git a/build/math.js b/build/math.js
@@ -4414,19 +4414,38 @@ function dirichletEta( x ) { return mul( zeta(x), sub( 1, pow( 2, sub(1,x) ) ) )
 
 function bernoulli( n, x ) {
 
-  if ( !Number.isInteger(n) ) throw Error( 'Noninteger index for Bernoulli number' );
+  if ( arguments.length === 2 ) {
 
-  if ( n < 0 ) throw Error( 'Unsupported index for Bernoulli number' );
+    if ( isZero(n) ) return isComplex(n) || isComplex(x) ? complex(1) : 1;
 
-  if ( arguments.length > 1 && !isZero(x) ) return mul( -n, hurwitzZeta(1-n,x) );
+    // avoid Hurwitz zeta parameter poles
+    if (  isNegativeIntegerOrZero(x) ) return complexAverage( x => bernoulli(n,x), x );
 
-  if ( n === 0 ) return 1;
+    return mul( neg(n), hurwitzZeta( sub(1,n), x ) );
+
+  }
+
+  if ( Number.isInteger(n) && n >= 0 ) {
+
+    if ( n === 0 ) return 1;
 
-  if ( n === 1 ) return -.5;
+    if ( n === 1 ) return -.5;
+
+    if ( n & 1 ) return 0;
+
+    var m = n/2;
+    if ( m <= bernoulli2nN.length )
+      return arbitrary( div( bernoulli2nN[m], bernoulli2nD[m] ) );
+
+    return -n * zeta(1-n);
+
+  }
 
-  if ( n & 1 ) return 0;
+  if ( isPositiveIntegerOrZero(n) ) return complex( bernoulli( n.re ) );
 
-  return -n * zeta(1-n);
+  // generalized Bernoulli number
+  console.log( 'Returning generalized Bernoulli number' );
+  return bernoulli( n, 0 );
 
 }
 

diff --git a/docs/functions.html b/docs/functions.html
@@ -335,9 +335,9 @@ <h1>Zeta Functions</h1>
 
 <p><a href="functions/dirichletEta.html"><b>dirichletEta( <i>x</i> )</b></a> &mdash; Dirichlet eta function of a real or complex number</p>
 
-<p><b>bernoulli( <i>n</i> )</b> &mdash; Bernoulli number for index <i>n</i></p>
+<p><b>bernoulli( <i>n</i> )</b> &mdash; Bernoulli number for index <i>n</i>. Returns a generalized value for fractional, negative or complex indices.</p>
 
-<p><b>bernoulli( <i>n</i>, <i>x</i> )</b> &mdash; Bernoulli polynomial for index <i>n</i> of a real or complex number</p>
+<p><b>bernoulli( <i>n</i>, <i>x</i> )</b> &mdash; Bernoulli polynomial for real or complex index <i>n</i> of a real or complex number</p>
 
 <p><b>harmonic( <i>n</i> )</b> &mdash; harmonic number for index <i>n</i></p>
 

diff --git a/src/functions/zeta.js b/src/functions/zeta.js
@@ -79,19 +79,38 @@ function dirichletEta( x ) { return mul( zeta(x), sub( 1, pow( 2, sub(1,x) ) ) )
 
 function bernoulli( n, x ) {
 
-  if ( !Number.isInteger(n) ) throw Error( 'Noninteger index for Bernoulli number' );
+  if ( arguments.length === 2 ) {
 
-  if ( n < 0 ) throw Error( 'Unsupported index for Bernoulli number' );
+    if ( isZero(n) ) return isComplex(n) || isComplex(x) ? complex(1) : 1;
 
-  if ( arguments.length > 1 && !isZero(x) ) return mul( -n, hurwitzZeta(1-n,x) );
+    // avoid Hurwitz zeta parameter poles
+    if (  isNegativeIntegerOrZero(x) ) return complexAverage( x => bernoulli(n,x), x );
 
-  if ( n === 0 ) return 1;
+    return mul( neg(n), hurwitzZeta( sub(1,n), x ) );
 
-  if ( n === 1 ) return -.5;
+  }
+
+  if ( Number.isInteger(n) && n >= 0 ) {
+
+    if ( n === 0 ) return 1;
+
+    if ( n === 1 ) return -.5;
+
+    if ( n & 1 ) return 0;
+
+    var m = n/2;
+    if ( m <= bernoulli2nN.length )
+      return arbitrary( div( bernoulli2nN[m], bernoulli2nD[m] ) );
+
+    return -n * zeta(1-n);
+
+  }
 
-  if ( n & 1 ) return 0;
+  if ( isPositiveIntegerOrZero(n) ) return complex( bernoulli( n.re ) );
 
-  return -n * zeta(1-n);
+  // generalized Bernoulli number
+  console.log( 'Returning generalized Bernoulli number' );
+  return bernoulli( n, 0 );
 
 }