BN

BN (LET)

Library Function

Compute the Bernoulli number or the Bernoulli polynomial.

The Bernoulli numbers can be defined by the recurrence relation:

where N defines the order of the Bernouilli numbers and "!" is the factorial function, and B(0) = 1.

The Bernoulli polynomials can be defined in terms of the Bernoulli numbers:

where BINOM is the binomial coefficient function (n!/(k!(n-k)!), and B(k) is the Bernoulli number of order k.

Dataplot computes this function using the BERNOB and BERNPN routines from "Computation of Special Functions" (see the Reference section below).

LET <y> = BN(<x>,<n>)             <SUBSET/EXCEPT/FOR qualification>
where <x> is a number, variable or parameter;
            <n> is a non-negagive integer number, variable or parameter;
            <y> is a variable or a parameter (depending on what <x> and <n> are) where the computed Bernoulli polynomial values are stored;
            and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes the Bernoulli polynomial of order .

LET <y> = BN(<n>)             <SUBSET/EXCEPT/FOR qualification>
where <n> is a non-negagive integer number, variable or parameter; and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes the Bernoulli number of order .

LET A = BN(12)
LET A = BN(B)
LET A = BN(2.5,4)
LET Y = BN(X,N)

None

None

BINCDF	= Compute the binomial cumulative distribution function.
BINPDF	= Compute the binomial probability mass function.
BINPPF	= Compute the binomial percent point function.
BINOMIAL	= Compute the binomial coefficients.
EN	= Compute Euler number or polynomial.

"Computation of Special Functions", Shanjie Zhang and Jianming Jin, John Wiley and Sons, 1996, chapter 1.

Probability

1997/12

TITLE BERNOULLI POLYNOMIAL OF ORDER 2
LET X = SEQUENCE 0 0.01 5
LET Y = BN(X,2)
RETAIN X Y SUBSET Y > 0
YLIMTIS 0 20
YTIC OFFSET 0 1
PLOT Y X