andragradspolynom

Andragradspolynom refers to a class of polynomials in one variable that are generated by a gradation-based three-term recurrence. In its standard form, the family {Pn(x)} over a field of characteristic zero is defined by P0(x) = 1, P1(x) = x, and, for n ≥ 1, Pn+1(x) = x·Pn(x) − αn·Pn−1(x), where (αn) is a prescribed real sequence called the gradation sequence. The degree of Pn is n, and the polynomials are typically studied for their structural and approximation properties arising from the recurrence.

Properties of andragradspolynom are closely linked to those of classical orthogonal polynomials. If the gradation sequence (αn) satisfies the conditions of Favard’s theorem, the polynomials form a sequence of polynomials orthogonal with respect to some positive measure on an interval of the real line. This orthogonality yields standard consequences such as a real, simple zero set contained in the interval of orthogonality, interlacing of zeros between consecutive polynomials, and a three-term recurrence relation characterizing the system.

Special cases and relationships: by selecting particular gradation sequences αn, and sometimes applying a scaling or shift of the variable x, andragradspolynom can reproduce or closely resemble classical families such as Hermite, Legendre, or Chebyshev-type polynomials. In general, the framework provides a way to construct new polynomial systems tailored to specific weight functions or domains, extending the toolbox of numerical approximation and spectral methods.

Applications and context: andragradspolynom are used in approximation theory, numerical analysis, and spectral methods where flexible polynomial bases are advantageous. They offer a unifying perspective for studying recurrence-based polynomial systems and their associated quadrature rules.

See also: orthogonal polynomials, Favard’s theorem, recurrence relations, quadrature.