NorlundB

The Nörlund polynomial $B_n^{(a)}$ , from the generating function $(t/(e^t-1))^a$ . At $a=1$ it is the ordinary Bernoulli number; in general, an exact polynomial in $a$ with bigint-rational coefficients, from a power-series log/exp of the Bernoulli EGF.

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NorlundB(n, a)

B_n^{(a)}

, exact.

Typed signatures

Type	Package	Applies
`(integer, number) -> number`	`analytic`	always

Domain: Special functions

Details

Computed by logging the EGF $t/(e^t-1) = \sum B_k t^k/k!$ into a power series $g(t)$ (the standard power-series-logarithm recurrence), then exponentiating $a \cdot g(t)$ back — a genuine polynomial identity in $a$ , since each convolution step contributes one more factor of $a$ . Every step is exact bigint-rational arithmetic; no float is involved until $a$ itself is one.
At a symbolic $a$ , returns the polynomial as a MathJSON expression in $a$ ; at a concrete rational $a$ , an exact rational number; at a float $a$ , a float.
Checked directly against Wolfram's own NorlundB at several $(n, a)$ pairs, including $n = 6, a = 4$ ( $221/42$ ).

Examples

3 more kept as data, checked against the oracles

Basic #

At a = 1 it is the Bernoulli number $B_2$

In

\mathrm{NorlundB}(2, 1)

Out

\frac{1}{6}

In

\mathrm{NorlundB}(4, 1)

Out

-\frac{1}{30}

In

\mathrm{NorlundB}(2, 2)

Out

\frac{5}{6}

In

\mathrm{NorlundB}(3, 2)

Out

-\frac{1}{2}

Scope #

A polynomial in a

In

\mathrm{NorlundB}(1, a)

Out

-\frac{a}{2}

Properties #

Zeroth-order is 1 regardless

In

\mathrm{NorlundB}(0, 0)

Out

1

Known value · Wolfram MathematicalFunctionData, NorlundB

With $a=0$ and $n=1$ it is 0

In

\mathrm{NorlundB}(1, 0)

Out

0

Known value · Wolfram MathematicalFunctionData, NorlundB

With $a=0$ and $n=2$ it is 0

In

\mathrm{NorlundB}(2, 0)

Out

0

Known value · Wolfram MathematicalFunctionData, NorlundB

Degree 0 always gives 1

In

\mathrm{NorlundB}(0, 2)

Out

1

Known value · Wolfram MathematicalFunctionData, NorlundB

$B_1^{(2)} = -1$

In

\mathrm{NorlundB}(1, 2)

Out

-1

Known value · Wolfram MathematicalFunctionData, NorlundB

Degree 0 for $a=3$

In

\mathrm{NorlundB}(0, 3)

Out

1

Known value · Wolfram MathematicalFunctionData, NorlundB

$B_1^{(a)}=-\frac a2$ , so $-\frac32$

In

\mathrm{NorlundB}(1, 3)

Out

-\frac{3}{2}

Known value · Wolfram MathematicalFunctionData, NorlundB

Degree 0 for $a=4$

In

\mathrm{NorlundB}(0, 4)

Out

1

Known value · Wolfram MathematicalFunctionData, NorlundB

$B_1^{(4)}=-2$

In

\mathrm{NorlundB}(1, 4)

Out

-2

Known value · Wolfram MathematicalFunctionData, NorlundB

Degree 0 for $a=5$

In

\mathrm{NorlundB}(0, 5)

Out

1

Known value · Wolfram MathematicalFunctionData, NorlundB

$B_1^{(5)}=-\frac52$

In

\mathrm{NorlundB}(1, 5)

Out

-\frac{5}{2}

Known value · Wolfram MathematicalFunctionData, NorlundB

Degree 0 for $a=6$

In

\mathrm{NorlundB}(0, 6)

Out

1

Known value · Wolfram MathematicalFunctionData, NorlundB

$B_1^{(6)}=-3$

In

\mathrm{NorlundB}(1, 6)

Out

-3

Known value · Wolfram MathematicalFunctionData, NorlundB

Degree 0 for $a=7$

In

\mathrm{NorlundB}(0, 7)

Out

1

Known value · Wolfram MathematicalFunctionData, NorlundB

$B_1^{(7)}=-\frac72$

In

\mathrm{NorlundB}(1, 7)

Out

-\frac{7}{2}

Known value · Wolfram MathematicalFunctionData, NorlundB

Degree 0 for $a=8$

In

\mathrm{NorlundB}(0, 8)

Out

1

Known value · Wolfram MathematicalFunctionData, NorlundB

$B_1^{(8)}=-4$

In

\mathrm{NorlundB}(1, 8)

Out

-4

Known value · Wolfram MathematicalFunctionData, NorlundB

About

Examples	25
Checked against	Wolfram Language 25 agree
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/norlund.ts

See also: BernoulliB