OpenMath Content Dictionary: orthog

Canonical URL:
http://www.openmath.org/CDs/orthog.ocd
CD File:
orthog.ocd
CD as XML Encoded OpenMath:
orthog.omcd
Defines:
ChebyshevT, ChebyshevU, GegenbauerC, HermiteH, LaguerreL
Date:
2001-26-08
Version:
1
Review Date:
Status:
experimental
Uses CD:
alg1, arith1, logic1, relation1

2002-01-01

This content dictionary contains symbols to describe the orthogonal polynomials. They are defined as in Abromowitz and Stegun (ninth printing on).


GegenbauerC

The GegenbauerC symbol represents a function to construct the ultraspherical (Gegenbauer) polynomials. it takes 3 arguments and is defined as follows:

If the first parameter is a non-negative integer, then the function GegenbauerC computes the nth ultraspherical (Gegenbauer) polynomial, with parameter a, evaluated at x.

These polynomials are orthogonal on the interval [-1,1], with respect to the weight function w(x) = (1-x^2)^(a-1/2). Thus

$\int^1_{-1}w(t)GegenbauerC(n, t) GegenbauerC(m, t) dt = 0$

if m is not equal to n.

The next five properties define recurence relations for these polynomials
    
Commented Mathematical property (CMP):
GegenbauerC(0,a,x) = 1
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA>
        <OMS cd="relation1" name="eq"/>
	<OMA><OMS cd="orthog" name="GegenbauerC"/>
	  <OMS cd="alg1" name="zero"/>
	  <OMV name="a"/>
	  <OMV name="x"/>
	</OMA>
	<OMS cd="alg1" name="one"/>
      </OMA>
    </OMOBJ>

eq (GegenbauerC (zero, a, x) , one)

Commented Mathematical property (CMP):
GegenbauerC(1,a,x) = 2*a*x
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA>
        <OMS cd="relation1" name="eq"/>
	<OMA><OMS cd="orthog" name="GegenbauerC"/>
	  <OMS cd="alg1" name="one"/>
	  <OMV name="a"/>
	  <OMV name="x"/>
	</OMA>
	<OMA><OMS cd="arith1" name="times"/>
	  <OMI>2</OMI>
	  <OMV name="a"/>
	  <OMV name="x"/>
	</OMA>
      </OMA>
    </OMOBJ>

eq (GegenbauerC (one, a, x) , times (2, a, x) )

Commented Mathematical property (CMP):
GegenbauerC(n,a,x) = 2*(n+a-1)/n*x*GegenbauerC(n-1,a,x) - (n+2*a-2)/n*GegenbauerC(n-2,a,x), for n gt 1
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA><OMS cd="logic1" name="implies"/>
        <OMA>
	  <OMS cd="relation1" name="gt"/>
	  <OMV name="n"/>
	  <OMS cd="alg1" name="one"/>
	</OMA>
	<OMA><OMS cd="relation1" name="eq"/>
	  <OMA><OMS cd="orthog" name="GegenbauerC"/>
	    <OMV name="n"/>
	    <OMV name="a"/>
	    <OMV name="x"/>
	  </OMA>
	  <OMA><OMS cd="arith1" name="minus"/>
	    <OMA><OMS cd="arith1" name="times"/>
	      <OMI>2</OMI>
	      <OMA><OMS cd="arith1" name="divide"/>
	        <OMA><OMS cd="arith1" name="plus"/>
		  <OMV name="n"/>
		  <OMA><OMS cd="arith1" name="minus"/>
		    <OMV name="a"/>
		    <OMS cd="alg1" name="one"/>
		  </OMA>
		</OMA>
		<OMV name="n"/>
	      </OMA>
	    </OMA>
	    <OMA><OMS cd="arith1" name="times"/>
	      <OMA><OMS cd="arith1" name="divide"/>
	        <OMA><OMS cd="arith1" name="plus"/>
		  <OMV name="n"/>
		  <OMA><OMS cd="arith1" name="minus"/>
		    <OMA><OMS cd="arith1" name="times"/>
		      <OMI>2</OMI>
		      <OMV name="a"/>
		    </OMA>
		    <OMI>2</OMI>
		  </OMA>
		</OMA>
		<OMV name="n"/>
	      </OMA>
	      <OMA><OMS cd="orthog" name="GegenbauerC"/>
	        <OMA><OMS cd="arith1" name="minus"/>
		  <OMV name="n"/>
		  <OMI>2</OMI>
		</OMA>
		<OMV name="a"/>
		<OMV name="x"/>
	      </OMA>
	    </OMA>
	  </OMA>
	</OMA>
      </OMA>
    </OMOBJ>

implies (gt ( n, one) , eq (GegenbauerC ( n, a, x) , minus (times (2, divide (plus ( n, minus ( a, one) ) , n) ) , times (divide (plus ( n, minus (times (2, a) , 2) ) , n) , GegenbauerC (minus ( n, 2) , a, x) ) ) ) )

Commented Mathematical property (CMP):
GegenbauerC(0,0,x) = 1
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA><OMS cd="relation1" name="eq"/>
	<OMA><OMS cd="orthog" name="GegenbauerC"/>
	  <OMS cd="alg1" name="zero"/>
	  <OMS cd="alg1" name="zero"/>
	  <OMV name="x"/>
	</OMA>
	<OMS cd="alg1" name="one"/>
      </OMA>
    </OMOBJ>

eq (GegenbauerC (zero, zero, x) , one)

Commented Mathematical property (CMP):
GegenbauerC(n,0,x) = 2/n*ChebyshevT(n,x) for n>0
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA><OMS cd="relation1" name="eq"/>
	<OMA><OMS cd="orthog" name="GegenbauerC"/>
	  <OMV name="n"/>
	  <OMS cd="alg1" name="zero"/>
	  <OMV name="x"/>
	</OMA>
	<OMA><OMS cd="arith1" name="times"/>
	  <OMA><OMS cd="arith1" name="divide"/>
	    <OMI>2</OMI>
	    <OMV name="n"/>
	  </OMA>
	  <OMA><OMS cd="orthog" name="ChebyshevT"/>
	    <OMV name="n"/>
	    <OMV name="x"/>
	  </OMA>
	</OMA>
      </OMA>
    </OMOBJ>

eq (GegenbauerC ( n, zero, x) , times (divide (2, n) , ChebyshevT ( n, x) ) )

Commented Mathematical property (CMP):
If the first parameter is not equal to a non-negative integer, then this is the analytic extension of the Gegenbauer polynomial and is such that: GegenbauerC(mu,nu,z) = GAMMA(mu+2*nu)/GAMMA(mu+1)/GAMMA(2*nu)* hypergeom([-mu,mu+2*nu],[nu+1/2],(1-z)/2);
Signatures:
sts


[Next: HermiteH] [Last: ChebyshevU] [Top]

HermiteH

The HermiteH symbol represents a function to construct the Hermite polynomials, it takes two parameters n, x. If the first parameter is a non-negative integer, the HermiteH(n, x) function computes the nth Hermite polynomial evaluated at x. The Hermite polynomials are orthogonal on the interval (-infinity, infinity), with respect to the weight function w(x) = exp(-x^2). Thus,

$\int^\infty_{-\infty}w(t) HermiteH(n, t) HermiteH(m, t)dt = 0$

if m is not equal to n.

The next three properties define recurence relations for these polynomials
    
Commented Mathematical property (CMP):
HermiteH(0,x) = 1
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA><OMS cd="relation1" name="eq"/>
        <OMA><OMS cd="orthog" name="HermiteH"/>
	  <OMS cd="alg1" name="zero"/>
	  <OMV name="x"/>
	</OMA>
	<OMS cd="alg1" name="one"/>
      </OMA>
    </OMOBJ>

eq (HermiteH (zero, x) , one)

Commented Mathematical property (CMP):
HermiteH(1,x) = 2*x
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA><OMS cd="relation1" name="eq"/>
        <OMA><OMS cd="orthog" name="HermiteH"/>
	  <OMS cd="alg1" name="one"/>
	  <OMV name="x"/>
	</OMA>
	<OMA><OMS cd="arith1" name="times"/>
	  <OMI>2</OMI>
	  <OMV name="x"/>
	</OMA>
      </OMA>
    </OMOBJ>

eq (HermiteH (one, x) , times (2, x) )

Commented Mathematical property (CMP):
HermiteH(n,x) = 2*x*HermiteH(n-1,x) - 2*(n-1)*HermiteH(n-2,x), for n gt 1
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA><OMS cd="logic1" name="implies"/>
        <OMA><OMS cd="relation1" name="gt"/>
	  <OMV name="n"/>
	  <OMS cd="alg1" name="one"/>
	</OMA>
	<OMA><OMS cd="relation1" name="eq"/>
	  <OMA><OMS cd="orthog" name="HermiteH"/>
	    <OMV name="n"/>
	    <OMV name="x"/>
	  </OMA>
	  <OMA><OMS cd="arith1" name="minus"/>
	    <OMA><OMS cd="arith1" name="times"/>
	      <OMI>2</OMI>
	      <OMV name="x"/>
	      <OMA><OMS cd="orthog" name="HermiteH"/>
	        <OMA><OMS cd="arith1" name="minus"/>
		  <OMV name="n"/>
		  <OMS cd="alg1" name="one"/>
		</OMA>
		<OMV name="x"/>
	      </OMA>
	    </OMA>
	    <OMA><OMS cd="arith1" name="times"/>
	      <OMI>2</OMI>
	      <OMA><OMS cd="arith1" name="minus"/>
	        <OMV name="n"/>
		<OMS cd="alg1" name="one"/>
	      </OMA>
	      <OMA><OMS cd="orthog" name="HermiteH"/>
	        <OMA><OMS cd="arith1" name="minus"/>
		  <OMV name="n"/>
		  <OMI>2</OMI>
		</OMA>
		<OMV name="x"/>
	      </OMA>
	    </OMA>
	  </OMA>
	</OMA>
      </OMA>
    </OMOBJ>

implies (gt ( n, one) , eq (HermiteH ( n, x) , minus (times (2, x, HermiteH (minus ( n, one) , x) ) , times (2, minus ( n, one) , HermiteH (minus ( n, 2) , x) ) ) ) )

Commented Mathematical property (CMP):
If the first parameter n is a negative integer, then this is the analytic extension of HermiteH and is such that: HermiteH(n,x) = 2^n*KummerU(-n/2,1/2,x^2) if Re(x) > 0
Signatures:
sts


[Next: LaguerreL] [Previous: GegenbauerC] [Top]

LaguerreL

The LaguerreL symbol represents a function to construct the Laguerre polynomials, it takes three parameters n, a and x. If a non-negative integer is entered as the first parameter, then the function LaguerreL will compute the nth generalized Laguerre polynomial with parameter a, evaluated at x. These polynomials are orthogonal on the interval [0,infinity), with respect to the weight function w(x) = exp(-x)*x^a. Thus

$\int^\infty_{0}w(t) LaguerreL(m, a, t) LaguerreL(n, a, t)dt = 0$

if m is not equal to n.

The next three properties define recurence relations for these polynomials
    
Commented Mathematical property (CMP):
LaguerreL(0,a,x) = 1
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA><OMS cd="relation1" name="eq"/>
        <OMA><OMS cd="orthog" name="LaguerreL"/>
	  <OMS cd="alg1" name="zero"/>
	  <OMV name="a"/>
	  <OMV name="x"/>
	</OMA>
	<OMS cd="alg1" name="one"/>
      </OMA>
    </OMOBJ>

eq (LaguerreL (zero, a, x) , one)

Commented Mathematical property (CMP):
LaguerreL(1,a,x) = -x+1+a
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA><OMS cd="relation1" name="eq"/>
        <OMA><OMS cd="orthog" name="LaguerreL"/>
	  <OMS cd="alg1" name="one"/>
	  <OMV name="a"/>
	  <OMV name="x"/>
	</OMA>
	<OMA><OMS cd="arith1" name="plus"/>
	  <OMA><OMS cd="arith1" name="unary_minus"/>
	    <OMV name="x"/>
	  </OMA>
	  <OMS cd="alg1" name="one"/>
	  <OMV name="a"/>
	</OMA>
      </OMA>
    </OMOBJ>

eq (LaguerreL (one, a, x) , plus (unary_minus ( x) , one, a) )

Commented Mathematical property (CMP):
LaguerreL(n,a,x) = (2*n+a-1-x)/n*LaguerreL(n-1,a,x) - (n+a-1)/n*LaguerreL(n-2,a,x) for n > 1.
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA><OMS cd="logic1" name="implies"/>
        <OMA><OMS cd="relation1" name="gt"/>
	  <OMV name="n"/>
	  <OMS cd="alg1" name="one"/>
	</OMA>
        <OMA><OMS cd="relation1" name="eq"/>
	  <OMA><OMS cd="orthog" name="LaguerreL"/>
	    <OMV name="n"/>
	    <OMV name="a"/>
	    <OMV name="x"/>
          </OMA>
	  <OMA><OMS cd="arith1" name="minus"/>
	    <OMA><OMS cd="arith1" name="times"/>
	      <OMA><OMS cd="arith1" name="divide"/>
	        <OMA><OMS cd="arith1" name="plus"/>
	          <OMA><OMS cd="arith1" name="times"/>
		    <OMI>2</OMI>
		    <OMV name="x"/>
		  </OMA>
		  <OMA><OMS cd="arith1" name="minus"/>
		    <OMA><OMS cd="arith1" name="minus"/>
		      <OMV name="a"/>
		      <OMS cd="alg1" name="one"/>
		    </OMA>
		    <OMV name="x"/>
		  </OMA>
	        </OMA>
		<OMV name="n"/>
	      </OMA>
	      <OMA><OMS cd="orthog" name="LaguerreL"/>
	        <OMA><OMS cd="arith1" name="minus"/>
		  <OMV name="n"/>
		  <OMS cd="alg1" name="one"/>
	        </OMA>
		<OMV name="a"/>
		<OMV name="x"/>
	      </OMA>
	    </OMA>
	    <OMA><OMS cd="arith1" name="times"/>
	      <OMA><OMS cd="arith1" name="divide"/>
	        <OMA><OMS cd="arith1" name="plus"/>
		  <OMV name="n"/>
		  <OMA><OMS cd="arith1" name="minus"/>
		    <OMV name="a"/>
		    <OMS cd="alg1" name="one"/>
		  </OMA>
		</OMA>
		<OMV name="n"/>
	      </OMA>
	      <OMA><OMS cd="orthog" name="LaguerreL"/>
	        <OMA><OMS cd="arith1" name="minus"/>
		  <OMV name="n"/>
		  <OMI>2</OMI>
	        </OMA>
		<OMV name="a"/>
		<OMV name="x"/>
	      </OMA>
	    </OMA>
	  </OMA>
        </OMA>
      </OMA>
    </OMOBJ>

implies (gt ( n, one) , eq (LaguerreL ( n, a, x) , minus (times (divide (plus (times (2, x) , minus (minus ( a, one) , x) ) , n) , LaguerreL (minus ( n, one) , a, x) ) , times (divide (plus ( n, minus ( a, one) ) , n) , LaguerreL (minus ( n, 2) , a, x) ) ) ) )

Commented Mathematical property (CMP):
If the first parameter is not equal to a non-negative integer, then this is the analytic extension of the Laguerre Polynomial and is such that: LaguerreL(mu,z) = hypergeom([-mu],[1],z); and LaguerreL(mu,nu,z) = GAMMA(mu+nu+1)/GAMMA(mu+1)/GAMMA(nu+1) * hypergeom([-mu],[nu+1],z);
Signatures:
sts


[Next: ChebyshevT] [Previous: HermiteH] [Top]

ChebyshevT

The ChebyshevT symbol represents a function to construct the Chebyshev polynomials, it takes two parameters n, x. These polynomials are orthogonal on the interval [-1, 1] with respect to the weight function w(x) = (1-x^2)^(-1/2). Thus

$\int^1_{-1}w(t) ChebyshevT(m, t) ChebyshevT(n, t)dt = 0$

if m is not equal to n.

The next three properties define recurence relations for these polynomials
    
Commented Mathematical property (CMP):
ChebyshevT(0,x) = 1
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMA><OMS cd="orthog" name="ChebyshevT"/>
        <OMS cd="alg1" name="zero"/>
	<OMV name="x"/>
      </OMA>
      <OMS cd="alg1" name="one"/>
    </OMA>
    </OMOBJ>

eq (ChebyshevT (zero, x) , one)

Commented Mathematical property (CMP):
ChebyshevT(1,x) = x
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMA><OMS cd="orthog" name="ChebyshevT"/>
        <OMS cd="alg1" name="one"/>
	<OMV name="x"/>
      </OMA>
      <OMV name="x"/>
    </OMA>
    </OMOBJ>

eq (ChebyshevT (one, x) , x)

Commented Mathematical property (CMP):
ChebyshevT(n,x) = 2*x*ChebyshevT(n-1,x) - ChebyshevT(n-2,x), for n gt 1
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA><OMS cd="logic1" name="implies"/>
        <OMA><OMS cd="relation1" name="gt"/>
	  <OMV name="n"/>
	  <OMS cd="alg1" name="one"/>
	</OMA>
        <OMA><OMS cd="relation1" name="eq"/>
	  <OMA><OMS cd="orthog" name="ChebyshevT"/>
	    <OMV name="n"/>
	    <OMV name="x"/>
          </OMA>
	  <OMA><OMS cd="arith1" name="minus"/>
	    <OMA><OMS cd="arith1" name="times"/>
	      <OMI>2</OMI>
	      <OMV name="x"/>
	      <OMA><OMS cd="orthog" name="ChebyshevT"/>
	        <OMA><OMS cd="arith1" name="minus"/>
		  <OMV name="n"/>
		  <OMS cd="alg1" name="one"/>
		</OMA>
		<OMV name="x"/>
	      </OMA>
	    </OMA>
	    <OMA><OMS cd="orthog" name="ChebyshevT"/>
	      <OMA><OMS cd="arith1" name="minus"/>
	        <OMV name="n"/>
		<OMI>2</OMI>
	      </OMA>
	      <OMV name="x"/>
	    </OMA>
	  </OMA>
        </OMA>
      </OMA>
    </OMOBJ>

implies (gt ( n, one) , eq (ChebyshevT ( n, x) , minus (times (2, x, ChebyshevT (minus ( n, one) , x) ) , ChebyshevT (minus ( n, 2) , x) ) ) )

Commented Mathematical property (CMP):
If the first parameter is not equal to a non-negative integer, then this is the analytic extension of the ChebyshevT polynomial and is such that: ChebyshevT(mu,z) = hypergeom([-mu,mu],[1/2],(1-z)/2)
Signatures:
sts


[Next: ChebyshevU] [Previous: LaguerreL] [Top]

ChebyshevU

The ChebyshevU symbol represents a function to construct the Chebyshev polynomials, it takes two parameters n, x. If a non-negative integer is entered as the first parameter, then the function ChebyshevU will compute the nth Chebyshev polynomial of the second kind, evaluated at x. These polynomials are orthogonal on the interval [-1, 1], with respect to the weight function w(x) = (1-x^2)^(1/2). Thus

$\int^1_{-1}w(t) ChebyshevU(m, t) ChebyshevU(n, t)dt = 0$

if m is not equal to n.

The next three properties define recurence relations for these polynomials
    
Commented Mathematical property (CMP):
ChebyshevU(0,x) = 1
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMA><OMS cd="orthog" name="ChebyshevU"/>
        <OMS cd="alg1" name="zero"/>
	<OMV name="x"/>
      </OMA>
      <OMS cd="alg1" name="one"/>
    </OMA>
    </OMOBJ>

eq (ChebyshevU (zero, x) , one)

Commented Mathematical property (CMP):
ChebyshevU(1,x) = 2*x
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMA><OMS cd="orthog" name="ChebyshevU"/>
        <OMS cd="alg1" name="one"/>
	<OMV name="x"/>
      </OMA>
      <OMA><OMS cd="arith1" name="times"/>
        <OMI>2</OMI>
        <OMV name="x"/>
      </OMA>
    </OMA>
    </OMOBJ>

eq (ChebyshevU (one, x) , times (2, x) )

Commented Mathematical property (CMP):
ChebyshevU(n,x) = 2*x*ChebyshevU(n-1,x) - ChebyshevU(n-2,x), for n gt 1
Formal Mathematical property (FMP):
<OMOBJ>
      <OMA><OMS cd="logic1" name="implies"/>
        <OMA><OMS cd="relation1" name="gt"/>
	  <OMV name="n"/>
	  <OMS cd="alg1" name="one"/>
	</OMA>
        <OMA><OMS cd="relation1" name="eq"/>
	  <OMA><OMS cd="orthog" name="ChebyshevU"/>
	    <OMV name="n"/>
	    <OMV name="x"/>
          </OMA>
	  <OMA><OMS cd="arith1" name="minus"/>
	    <OMA><OMS cd="arith1" name="times"/>
	      <OMI>2</OMI>
	      <OMV name="x"/>
	      <OMA><OMS cd="orthog" name="ChebyshevU"/>
	        <OMA><OMS cd="arith1" name="minus"/>
		  <OMV name="n"/>
		  <OMS cd="alg1" name="one"/>
		</OMA>
		<OMV name="x"/>
	      </OMA>
	    </OMA>
	    <OMA><OMS cd="orthog" name="ChebyshevU"/>
	      <OMA><OMS cd="arith1" name="minus"/>
	        <OMV name="n"/>
		<OMI>2</OMI>
	      </OMA>
	      <OMV name="x"/>
	    </OMA>
	  </OMA>
        </OMA>
      </OMA>
    </OMOBJ>

implies (gt ( n, one) , eq (ChebyshevU ( n, x) , minus (times (2, x, ChebyshevU (minus ( n, one) , x) ) , ChebyshevU (minus ( n, 2) , x) ) ) )

Commented Mathematical property (CMP):
If the first parameter is not equal to a non-negative integer, then this is the analytic extension of ChebyshevU and is such that: ChebyshevU(mu,z) = hypergeom([-mu,mu+2],[3/2],(1-z)/2)
Signatures:
sts


[First: GegenbauerC] [Previous: ChebyshevT] [Top]