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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/algebra infprod.spad}
+\author{Clifton J. Williamson}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{package STINPROD StreamInfiniteProduct}
+<<package STINPROD StreamInfiniteProduct>>=
+)abbrev package STINPROD StreamInfiniteProduct
+++ Author: Clifton J. Williamson
+++ Date Created: 23 February 1990
+++ Date Last Updated: 23 February 1990
+++ Basic Operations: infiniteProduct, evenInfiniteProduct, oddInfiniteProduct,
+++   generalInfiniteProduct
+++ Related Domains: UnivariateTaylorSeriesCategory
+++ Also See:
+++ AMS Classifications:
+++ Keywords: Taylor series, infinite product
+++ Examples:
+++ References:
+++ Description: 
+++   This package computes infinite products of Taylor series over an
+++   integral domain of characteristic 0.  Here Taylor series are
+++   represented by streams of Taylor coefficients.
+StreamInfiniteProduct(Coef): Exports == Implementation where
+  Coef: Join(IntegralDomain,CharacteristicZero)
+  I  ==> Integer
+  QF ==> Fraction
+  ST ==> Stream
+ 
+  Exports ==> with
+ 
+    infiniteProduct: ST Coef -> ST Coef
+      ++ infiniteProduct(f(x)) computes \spad{product(n=1,2,3...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+    evenInfiniteProduct: ST Coef -> ST Coef
+      ++ evenInfiniteProduct(f(x)) computes \spad{product(n=2,4,6...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+    oddInfiniteProduct: ST Coef -> ST Coef
+      ++ oddInfiniteProduct(f(x)) computes \spad{product(n=1,3,5...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+    generalInfiniteProduct: (ST Coef,I,I) -> ST Coef
+      ++ generalInfiniteProduct(f(x),a,d) computes
+      ++ \spad{product(n=a,a+d,a+2*d,...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+ 
+  Implementation ==> add
+ 
+    if Coef has Field then
+ 
+      import StreamTaylorSeriesOperations(Coef)
+      import StreamTranscendentalFunctions(Coef)
+ 
+      infiniteProduct st             == exp lambert log st
+      evenInfiniteProduct st         == exp evenlambert log st
+      oddInfiniteProduct st          == exp oddlambert log st
+      generalInfiniteProduct(st,a,d) == exp generalLambert(log st,a,d)
+ 
+    else
+ 
+      import StreamTaylorSeriesOperations(QF Coef)
+      import StreamTranscendentalFunctions(QF Coef)
+ 
+      applyOverQF:(ST QF Coef -> ST QF Coef,ST Coef) -> ST Coef
+      applyOverQF(f,st) ==
+        stQF := map(#1 :: QF(Coef),st)$StreamFunctions2(Coef,QF Coef)
+        map(retract(#1)@Coef,f stQF)$StreamFunctions2(QF Coef,Coef)
+ 
+      infiniteProduct st     == applyOverQF(exp lambert log #1,st)
+      evenInfiniteProduct st == applyOverQF(exp evenlambert log #1,st)
+      oddInfiniteProduct st  == applyOverQF(exp oddlambert log #1,st)
+      generalInfiniteProduct(st,a,d) ==
+        applyOverQF(exp generalLambert(log #1,a,d),st)
+
+@
+\section{package INFPROD0 InfiniteProductCharacteristicZero}
+<<package INFPROD0 InfiniteProductCharacteristicZero>>=
+)abbrev package INFPROD0 InfiniteProductCharacteristicZero
+++ Author: Clifton J. Williamson
+++ Date Created: 22 February 1990
+++ Date Last Updated: 23 February 1990
+++ Basic Operations: infiniteProduct, evenInfiniteProduct, oddInfiniteProduct,
+++   generalInfiniteProduct
+++ Related Domains: UnivariateTaylorSeriesCategory
+++ Also See:
+++ AMS Classifications:
+++ Keywords: Taylor series, infinite product
+++ Examples:
+++ References:
+++ Description: 
+++   This package computes infinite products of univariate Taylor series
+++   over an integral domain of characteristic 0.
+InfiniteProductCharacteristicZero(Coef,UTS):_
+ Exports == Implementation where
+  Coef : Join(IntegralDomain,CharacteristicZero)
+  UTS  : UnivariateTaylorSeriesCategory Coef
+  I  ==> Integer
+ 
+  Exports ==> with
+ 
+    infiniteProduct: UTS -> UTS
+      ++ infiniteProduct(f(x)) computes \spad{product(n=1,2,3...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+    evenInfiniteProduct: UTS -> UTS
+      ++ evenInfiniteProduct(f(x)) computes \spad{product(n=2,4,6...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+    oddInfiniteProduct: UTS -> UTS
+      ++ oddInfiniteProduct(f(x)) computes \spad{product(n=1,3,5...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+    generalInfiniteProduct: (UTS,I,I) -> UTS
+      ++ generalInfiniteProduct(f(x),a,d) computes
+      ++ \spad{product(n=a,a+d,a+2*d,...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+ 
+  Implementation ==> add
+ 
+    import StreamInfiniteProduct Coef
+ 
+    infiniteProduct x     == series infiniteProduct coefficients x
+    evenInfiniteProduct x == series evenInfiniteProduct coefficients x
+    oddInfiniteProduct x  == series oddInfiniteProduct coefficients x
+ 
+    generalInfiniteProduct(x,a,d) ==
+      series generalInfiniteProduct(coefficients x,a,d)
+
+@
+\section{package INPRODPF InfiniteProductPrimeField}
+<<package INPRODPF InfiniteProductPrimeField>>=
+)abbrev package INPRODPF InfiniteProductPrimeField
+++ Author: Clifton J. Williamson
+++ Date Created: 22 February 1990
+++ Date Last Updated: 23 February 1990
+++ Basic Operations: infiniteProduct, evenInfiniteProduct, oddInfiniteProduct,
+++   generalInfiniteProduct
+++ Related Domains: UnivariateTaylorSeriesCategory
+++ Also See:
+++ AMS Classifications:
+++ Keywords: Taylor series, infinite product
+++ Examples:
+++ References:
+++ Description: 
+++    This package computes infinite products of univariate Taylor series
+++    over a field of prime order.
+InfiniteProductPrimeField(Coef,UTS): Exports == Implementation where
+  Coef : Join(Field,Finite,ConvertibleTo Integer)
+  UTS  : UnivariateTaylorSeriesCategory Coef
+  I  ==> Integer
+  ST ==> Stream
+ 
+  Exports ==> with
+ 
+    infiniteProduct: UTS -> UTS
+      ++ infiniteProduct(f(x)) computes \spad{product(n=1,2,3...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+    evenInfiniteProduct: UTS -> UTS
+      ++ evenInfiniteProduct(f(x)) computes \spad{product(n=2,4,6...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+    oddInfiniteProduct: UTS -> UTS
+      ++ oddInfiniteProduct(f(x)) computes \spad{product(n=1,3,5...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+    generalInfiniteProduct: (UTS,I,I) -> UTS
+      ++ generalInfiniteProduct(f(x),a,d) computes
+      ++ \spad{product(n=a,a+d,a+2*d,...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+ 
+  Implementation ==> add
+ 
+    import StreamInfiniteProduct Integer
+ 
+    applyOverZ:(ST I -> ST I,ST Coef) -> ST Coef
+    applyOverZ(f,st) ==
+      stZ := map(convert(#1)@Integer,st)$StreamFunctions2(Coef,I)
+      map(#1 :: Coef,f stZ)$StreamFunctions2(I,Coef)
+ 
+    infiniteProduct x ==
+      series applyOverZ(infiniteProduct,coefficients x)
+    evenInfiniteProduct x ==
+      series applyOverZ(evenInfiniteProduct,coefficients x)
+    oddInfiniteProduct x ==
+      series applyOverZ(oddInfiniteProduct,coefficients x)
+    generalInfiniteProduct(x,a,d) ==
+      series applyOverZ(generalInfiniteProduct(#1,a,d),coefficients x)
+
+@
+\section{package INPRODFF InfiniteProductFiniteField}
+<<package INPRODFF InfiniteProductFiniteField>>=
+)abbrev package INPRODFF InfiniteProductFiniteField
+++ Author: Clifton J. Williamson
+++ Date Created: 22 February 1990
+++ Date Last Updated: 23 February 1990
+++ Basic Operations: infiniteProduct, evenInfiniteProduct, oddInfiniteProduct,
+++   generalInfiniteProduct
+++ Related Domains: UnivariateTaylorSeriesCategory
+++ Also See:
+++ AMS Classifications:
+++ Keywords: Taylor series, infinite product
+++ Examples:
+++ References:
+++ Description: 
+++   This package computes infinite products of univariate Taylor series
+++   over an arbitrary finite field.
+InfiniteProductFiniteField(K,UP,Coef,UTS):_
+ Exports == Implementation where
+  K    :  Join(Field,Finite,ConvertibleTo Integer)
+  UP   :  UnivariatePolynomialCategory K
+  Coef :  MonogenicAlgebra(K,UP)
+  UTS  :  UnivariateTaylorSeriesCategory Coef
+  I   ==> Integer
+  RN  ==> Fraction Integer
+  SAE ==> SimpleAlgebraicExtension
+  ST  ==> Stream
+  STF ==> StreamTranscendentalFunctions
+  STT ==> StreamTaylorSeriesOperations
+  ST2 ==> StreamFunctions2
+  SUP ==> SparseUnivariatePolynomial
+ 
+  Exports ==> with
+ 
+    infiniteProduct: UTS -> UTS
+      ++ infiniteProduct(f(x)) computes \spad{product(n=1,2,3...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+    evenInfiniteProduct: UTS -> UTS
+      ++ evenInfiniteProduct(f(x)) computes \spad{product(n=2,4,6...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+    oddInfiniteProduct: UTS -> UTS
+      ++ oddInfiniteProduct(f(x)) computes \spad{product(n=1,3,5...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+    generalInfiniteProduct: (UTS,I,I) -> UTS
+      ++ generalInfiniteProduct(f(x),a,d) computes
+      ++ \spad{product(n=a,a+d,a+2*d,...,f(x**n))}.
+      ++ The series \spad{f(x)} should have constant coefficient 1.
+ 
+  Implementation ==> add
+ 
+    liftPoly: UP -> SUP RN
+    liftPoly poly ==
+      -- lift coefficients of 'poly' to integers
+      ans : SUP RN := 0
+      while not zero? poly repeat
+        coef := convert(leadingCoefficient poly)@I :: RN
+        ans := ans + monomial(coef,degree poly)
+        poly := reductum poly
+      ans
+ 
+    reducePoly: SUP RN -> UP
+    reducePoly poly ==
+      -- reduce coefficients of 'poly' to elements of K
+      ans : UP := 0
+      while not zero? poly repeat
+        coef := numer(leadingCoefficient(poly)) :: K
+        ans := ans + monomial(coef,degree poly)
+        poly := reductum poly
+      ans
+ 
+    POLY := liftPoly definingPolynomial()$Coef
+    ALG  := SAE(RN,SUP RN,POLY)
+ 
+    infiniteProduct x ==
+      stUP := map(lift,coefficients x)$ST2(Coef,UP)
+      stSUP := map(liftPoly,stUP)$ST2(UP,SUP RN)
+      stALG := map(reduce,stSUP)$ST2(SUP RN,ALG)
+      stALG := exp(lambert(log(stALG)$STF(ALG))$STT(ALG))$STF(ALG)
+      stSUP := map(lift,stALG)$ST2(ALG,SUP RN)
+      stUP := map(reducePoly,stSUP)$ST2(SUP RN,UP)
+      series map(reduce,stUP)$ST2(UP,Coef)
+ 
+    evenInfiniteProduct x ==
+      stUP := map(lift,coefficients x)$ST2(Coef,UP)
+      stSUP := map(liftPoly,stUP)$ST2(UP,SUP RN)
+      stALG := map(reduce,stSUP)$ST2(SUP RN,ALG)
+      stALG := exp(evenlambert(log(stALG)$STF(ALG))$STT(ALG))$STF(ALG)
+      stSUP := map(lift,stALG)$ST2(ALG,SUP RN)
+      stUP := map(reducePoly,stSUP)$ST2(SUP RN,UP)
+      series map(reduce,stUP)$ST2(UP,Coef)
+ 
+    oddInfiniteProduct x ==
+      stUP := map(lift,coefficients x)$ST2(Coef,UP)
+      stSUP := map(liftPoly,stUP)$ST2(UP,SUP RN)
+      stALG := map(reduce,stSUP)$ST2(SUP RN,ALG)
+      stALG := exp(oddlambert(log(stALG)$STF(ALG))$STT(ALG))$STF(ALG)
+      stSUP := map(lift,stALG)$ST2(ALG,SUP RN)
+      stUP := map(reducePoly,stSUP)$ST2(SUP RN,UP)
+      series map(reduce,stUP)$ST2(UP,Coef)
+ 
+    generalInfiniteProduct(x,a,d) ==
+      stUP := map(lift,coefficients x)$ST2(Coef,UP)
+      stSUP := map(liftPoly,stUP)$ST2(UP,SUP RN)
+      stALG := map(reduce,stSUP)$ST2(SUP RN,ALG)
+      stALG := generalLambert(log(stALG)$STF(ALG),a,d)$STT(ALG)
+      stALG := exp(stALG)$STF(ALG)
+      stSUP := map(lift,stALG)$ST2(ALG,SUP RN)
+      stUP := map(reducePoly,stSUP)$ST2(SUP RN,UP)
+      series map(reduce,stUP)$ST2(UP,Coef)
+
+@
+\section{License}
+<<license>>=
+--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
+--All rights reserved.
+--
+--Redistribution and use in source and binary forms, with or without
+--modification, are permitted provided that the following conditions are
+--met:
+--
+--    - Redistributions of source code must retain the above copyright
+--      notice, this list of conditions and the following disclaimer.
+--
+--    - Redistributions in binary form must reproduce the above copyright
+--      notice, this list of conditions and the following disclaimer in
+--      the documentation and/or other materials provided with the
+--      distribution.
+--
+--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
+--      names of its contributors may be used to endorse or promote products
+--      derived from this software without specific prior written permission.
+--
+--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
+--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
+--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
+--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
+--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
+--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+@
+<<*>>=
+<<license>>
+ 
+<<package STINPROD StreamInfiniteProduct>>
+<<package INFPROD0 InfiniteProductCharacteristicZero>>
+<<package INPRODPF InfiniteProductPrimeField>>
+<<package INPRODFF InfiniteProductFiniteField>>
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}