Carl M Bender and E Ben-Naim 2007 J. Phys. A: Math. Theor. 40 F9 doi:10.1088/1751-8113/40/1/F02
Carl M Bender1 and E Ben-Naim
Show affiliationsThe nonlinear integral equation P(x) = ∫βα dy w(y)P(y)P(x + y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions P(x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations for the coefficients of the polynomials. Interestingly, the set of polynomial solutions is orthogonal with respect to the measure xw(x). The nonlinear integral equation can be used to specify all orthogonal polynomials in a simple and compact way. This integral equation provides a natural vehicle for extending the theory of orthogonal polynomials into the complex domain. Generalizations of the integral equation are discussed.
Issue 1 (5 January 2007)
Received 26 October 2006, in final form 15 November 2006
Published 6 December 2006
Carl M Bender and E Ben-Naim 2007 J. Phys. A: Math. Theor. 40 F9
Frédéric Noo et al 2003 Phys. Med. Biol. 48 3787
Masayuki Abe et al 2005 Nanotechnology 16 3029
Jan W T Heemskerk et al 2009 Phys. Med. Biol. 54 3003
Weiguo Lu et al 2004 Phys. Med. Biol. 49 3067
W T Hung et al 1994 Phys. Med. Biol. 39 1855
Noriaki Oyabu et al 2005 Nanotechnology 16 S112
Gregory C Sharp et al 2004 Phys. Med. Biol. 49 425
H P Lawrence 1990 Phys. Med. Biol. 35 787
X X Xi et al 2002 Supercond. Sci. Technol. 15 451