Nothing
gegenbauer.recurrences <- function( n, alpha, normalized=FALSE )
{
###
### This function returns a data frame with n+1 and four columns
### containing the coefficients c, d, e and f of the recurrence relations
### for the order k generalized Laguerre polynomials, Lk(alpha,x),
### and for orders k=0,1,...n
###
### Parameters
### n = integer highest polynomial order
### alpha = polynomial parameter
### normalized = a boolean value. If true, recurrences are for normalized polynomials
###
if ( n < 0 )
stop( "negative highest polynomial order" )
if ( n != round( n ) )
stop( "highest polynomial order is not integer" )
if ( alpha <= -0.5 )
stop( "alpha less than or equal to -0.5" )
###
### alpha = 1.0
### special case is the Chebyshev polynomial of
###the second kind U_k(x)
###
if ( abs( alpha - 1 ) < 1e-6 )
return( chebyshev.u.recurrences( n, normalized ) )
###
### alpha = 0.5
### special case is the Legendre polynomial P_k(x)
###
if ( abs( alpha - 0.5 ) < 1e-6 )
return( legendre.recurrences( n, normalized ) )
np1 <- n + 1
r <- data.frame( matrix( nrow=np1, ncol=4 ) )
names( r ) <- c( "c", "d", "e", "f" )
j <- 0
k <- 1
###
### alpha = 0,0
### special case is related to the Chebyshev polynomial of
### the first kind T_k(x)
###
if ( abs( alpha ) < 1e-6 ) {
if ( normalized ) {
while( j <= n ) {
r[k,"c"] <- j + 1
r[k,"d"] <- 0
if ( j == 0 ) {
r[k,"e"] <- sqrt( 2 )
}
else {
r[k,"e"] <- 2 * ( j + 1 )
}
if ( j == 0 ) {
r[k,"f"] <- 0
}
else {
if ( j == 1 ) {
r[k,"f"] <- 2 * sqrt( 2 )
}
else {
r[k,"f"] <- j + 1
}
}
j <- j + 1
k <- k + 1
} # end while block
} # end if normalized block
else {
while ( j <= n ) {
r[k,"c"] <- j + 1
r[k,"d"] <- 0
if ( j == 0 ) {
r[k,"e"] <- 2
}
else {
r[k,"e"] <- 2 * j
}
if ( j == 0 ) {
r[k,"f"] <- 0
}
else {
if ( j == 1 ) {
r[k,"f"] <- 2
}
else {
r[k,"f"] <- j - 1
}
}
j <- j + 1
k <- k + 1
} # end while
} # end else not normalized block
return( r )
} # end if block
###
### general case
###
two.alpha <- 2 * alpha
if ( normalized ) {
while ( j <= n ) {
r[k,"c"] <- ( j + 1 )
r[k,"d"] <- 0
rho.j <- sqrt( ( alpha + j + 1 ) * ( j + 1 ) * gamma( two.alpha + j ) /
( ( alpha + j ) * gamma( two.alpha + j + 1 ) ) )
r[k,"e"] <- 2 * ( alpha + j ) * rho.j
if ( j == 0 ) {
rho.jm1 <- 0
}
else {
rho.jm1 <- sqrt( j * ( j + 1 ) * ( alpha + j + 1 ) * gamma( two.alpha + j - 1 ) /
( ( alpha + j - 1 ) * gamma( two.alpha + j + 1 ) ) )
}
r[k,"f"] <- ( j + 2 * alpha - 1 ) * rho.jm1
j <- j + 1
k <- k + 1
} # end while block
} # end if block
else {
while ( j <= n ) {
r[k,"c"] <- j + 1
r[k,"d"] <- 0
r[k,"e"] <- 2 * ( j + alpha )
r[k,"f"] <- j + 2 * alpha - 1
j <- j + 1
k <- k + 1
} # end while block
} # end of else block
return( r )
}
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