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A new method of composition orthogonality is introduced. It is applied togenerate new sequences of orthogonal polynomials and functions. In particular,classical orthogonal polynomials are interpreted in the sense of compositionorthogonality. Finally, new sequences of orthogonal polynomials with respect tothe weight function $x^\alpha \rho^2_\nu(x), \rho_{\nu}(x)= 2 x^{\nu/2}K_\nu(2\sqrt x),\ x >0, \nu \ge 0, \alpha > -1$, where $K_\nu(z)$ is themodified Bessel function or Macdonald function, are investigated. Differentialproperties, recurrence relations, explicit representations, generatingfunctions and Rodrigues-type formulae are obtained. The corresponding multipleorthogonal polynomials are exhibited.

A method of composition orthogonality and new sequences of orthogonal polynomials and functions for non-classical weights