直交関数列
∫ − 1 1 ( 1 − x 2 ) α − 1 2 C m ( α ) ( x ) C n ( α ) ( x ) d x = π 2 1 − 2 α Γ ( n + 2 α ) n ! ( n + α ) [ Γ ( α ) ] 2 δ m n ( m , n = 0 , 1 , 2 , ... ) {\displaystyle \int _{-1}^{1}(1-x^{2})^{\alpha -{\frac {1}{2}}}C_{m}^{(\alpha )}(x)C_{n}^{(\alpha )}(x)\,dx={\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )[\Gamma (\alpha )]^{2}}}\delta _{mn}\quad (m,n=0,1,2,\dots )}
を満たす。