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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input psgenfcn.input}
\author{Clifton J. Williamson}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{License}
<<license>>=
--Copyright The Numerical Algorithms Group Limited 1991.
@
<<*>>=
<<license>>
--% Power Series Generating Functions
-- We obtain Fibonacci numbers, Bernoulli numbers and polynomials, and
-- Euler numbers and polynomials as power series coefficients
-- Author: Clifton J. Williamson
-- Date Created: 1 June 1990
-- Date Last Updated: 1 June 1990
-- Keywords: Fibonacci, Bernoulli, Euler
-- References:
-- We will compute power series in two different ways, then test equality
-- of coefficients up to a specified order. The function that tests
-- 'approximate equality' will return an error message if the test fails.
)cl all
ORD := 20
approximateEquality(series1,series2) ==
-- tests that 2 series are equal to order ORD
uts1 := series1 :: UTS(EXPR INT,'t,0)
uts2 := series2 :: UTS(EXPR INT,'t,0)
flag := (order(uts1 - uts2,ORD) = ORD) :: Boolean
flag => true
error "series do not agree to specified order"
bernoulliPolynomial(n) ==
-- returns the nth Bernoulli polynomial as an EXPR INT
sup := bernoulli(n)$(PNTHEORY)
p : POLY FRAC INT := multivariate(sup,'x)
p :: (EXPR INT)
eulerPolynomial(n) ==
-- returns the nth Euler polynomial as an EXPR INT
sup := euler(n)$(PNTHEORY)
p : POLY FRAC INT := multivariate(sup,'x)
p :: (EXPR INT)
f1 := taylor(t/(1 - t - t**2))
f2 := taylor(n +-> fibonacci(n),t = 0)
approximateEquality(f1,f2)
g1 := taylor(t/(exp(t) - 1))
g2 := taylor(n +-> bernoulli(n)/factorial(n),t = 0)
approximateEquality(g1,g2)
gg1 := taylor(t*exp(t*x)/(exp(t) - 1),t = 0)
gg2 := taylor(n +-> bernoulliPolynomial(n)/factorial(n),t = 0)
approximateEquality(gg1,gg2)
h1 := taylor(2*exp(t/2)/(exp(t) + 1))
h2 := taylor(n +-> euler(n)/(2**n * factorial(n)),t = 0)
approximateEquality(h1,h2)
hh1 := taylor(2*exp(t*x)/(exp(t) + 1),t = 0)
hh2 := taylor(n +-> eulerPolynomial(n)/factorial(n),t = 0)
approximateEquality(hh1,hh2)
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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