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Characterizations

A variety of characterizations of the Gegenbauer polynomials are available.

• The polynomials can be defined in terms of their generating function (Stein & Weiss 1971, §IV.2):

1 ( 1 − 2 x t + t 2 ) α = ∑ n = 0 ∞ C n ( α ) ( x ) t n ( 0 ≤ | x | < 1 , | t | ≤ 1 , α > 0 ) {\displaystyle {\frac {1}{(1-2xt+t^{2})^{\alpha }}}=\sum _{n=0}^{\infty }C_{n}^{(\alpha )}(x)t^{n}\qquad (0\leq |x|<1,|t|\leq 1,\alpha >0)}

• The polynomials satisfy the recurrence relation (Suetin 2001):

C 0 ( α ) ( x ) = 1 C 1 ( α ) ( x ) = 2 α x ( n + 1 ) C n + 1 ( α ) ( x ) = 2 ( n + α ) x C n ( α ) ( x ) − ( n + 2 α − 1 ) C n − 1 ( α ) ( x ) . {\displaystyle {\begin{aligned}C_{0}^{(\alpha )}(x)&=1\\C_{1}^{(\alpha )}(x)&=2\alpha x\\(n+1)C_{n+1}^{(\alpha )}(x)&=2(n+\alpha )xC_{n}^{(\alpha )}(x)-(n+2\alpha -1)C_{n-1}^{(\alpha )}(x).\end{aligned}}}

• Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (Suetin 2001):

( 1 − x 2 ) y ″ − ( 2 α + 1 ) x y ′ + n ( n + 2 α ) y = 0. {\displaystyle (1-x^{2})y''-(2\alpha +1)xy'+n(n+2\alpha )y=0.\,} When α = 1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the Legendre polynomials. When α = 1, the equation reduces to the Chebyshev differential equation, and the Gegenbauer polynomials reduce to the Chebyshev polynomials of the second kind.¹

• They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite:

C n ( α ) ( z ) = ( 2 α ) n n ! 2 F 1 ( − n , 2 α + n ; α + 1 2 ; 1 − z 2 ) . {\displaystyle C_{n}^{(\alpha )}(z)={\frac {(2\alpha )_{n}}{n!}}\,_{2}F_{1}\left(-n,2\alpha +n;\alpha +{\frac {1}{2}};{\frac {1-z}{2}}\right).} (Abramowitz & Stegun p. 561). Here (2α)n is the rising factorial. Explicitly, C n ( α ) ( z ) = ∑ k = 0 ⌊ n / 2 ⌋ ( − 1 ) k Γ ( n − k + α ) Γ ( α ) k ! ( n − 2 k ) ! ( 2 z ) n − 2 k . {\displaystyle C_{n}^{(\alpha )}(z)=\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{k}{\frac {\Gamma (n-k+\alpha )}{\Gamma (\alpha )k!(n-2k)!}}(2z)^{n-2k}.} From this it is also easy to obtain the value at unit argument: C n ( α ) ( 1 ) = Γ ( 2 α + n ) Γ ( 2 α ) n ! . {\displaystyle C_{n}^{(\alpha )}(1)={\frac {\Gamma (2\alpha +n)}{\Gamma (2\alpha )n!}}.}

• They are special cases of the Jacobi polynomials (Suetin 2001):

C n ( α ) ( x ) = ( 2 α ) n ( α + 1 2 ) n P n ( α − 1 / 2 , α − 1 / 2 ) ( x ) . {\displaystyle C_{n}^{(\alpha )}(x)={\frac {(2\alpha )_{n}}{(\alpha +{\frac {1}{2}})_{n}}}P_{n}^{(\alpha -1/2,\alpha -1/2)}(x).} in which ( θ ) n {\displaystyle (\theta )_{n}} represents the rising factorial of θ {\displaystyle \theta } . One therefore also has the Rodrigues formula C n ( α ) ( x ) = ( − 1 ) n 2 n n ! Γ ( α + 1 2 ) Γ ( n + 2 α ) Γ ( 2 α ) Γ ( α + n + 1 2 ) ( 1 − x 2 ) − α + 1 / 2 d n d x n [ ( 1 − x 2 ) n + α − 1 / 2 ] . {\displaystyle C_{n}^{(\alpha )}(x)={\frac {(-1)^{n}}{2^{n}n!}}{\frac {\Gamma (\alpha +{\frac {1}{2}})\Gamma (n+2\alpha )}{\Gamma (2\alpha )\Gamma (\alpha +n+{\frac {1}{2}})}}(1-x^{2})^{-\alpha +1/2}{\frac {d^{n}}{dx^{n}}}\left[(1-x^{2})^{n+\alpha -1/2}\right].}

• An alternative normalization sets C n ( α ) ( 1 ) = 1 {\displaystyle C_{n}^{(\alpha )}(1)=1} . Assuming this alternative normalization, the derivatives of Gegenbauer are expressed in terms of Gegenbauer:²

d p d x p C q + 2 j + 1 ( α ) ( x ) = 2 q ( q + 2 j + 1 ) ! ( q − 1 ) ! Γ ( q + 2 j + 2 α + 1 ) ∑ i = 0 j ( 2 i + α + 1 ) Γ ( 2 i + 2 α + 1 ) ( 2 i + 1 ) ! ( j − i ) ! × Γ ( q + j + i + α + 1 ) Γ ( j + i + α + 2 ) ( q + j − i − 1 ) ! C 2 i + 1 ( α ) ( x ) {\displaystyle {\begin{aligned}{\frac {d^{p}}{dx^{p}}}C_{q+2j+1}^{(\alpha )}(x)={\frac {2q(q+2j+1)!}{(q-1)!\Gamma (q+2j+2\alpha +1)}}&\sum _{i=0}^{j}{\frac {(2i+\alpha +1)\Gamma (2i+2\alpha +1)}{(2i+1)!(j-i)!}}\\&\times {\frac {\Gamma (q+j+i+\alpha +1)}{\Gamma (j+i+\alpha +2)}}(q+j-i-1)!C_{2i+1}^{(\alpha )}(x)\end{aligned}}}

Orthogonality and normalization

For a fixed α > -1/2, the polynomials are orthogonal on [−1, 1] with respect to the weighting function (Abramowitz & Stegun p. 774)

w ( z ) = ( 1 − z 2 ) α − 1 2 . {\displaystyle w(z)=\left(1-z^{2}\right)^{\alpha -{\frac {1}{2}}}.}

To wit, for n ≠ m,

∫ − 1 1 C n ( α ) ( x ) C m ( α ) ( x ) ( 1 − x 2 ) α − 1 2 d x = 0. {\displaystyle \int _{-1}^{1}C_{n}^{(\alpha )}(x)C_{m}^{(\alpha )}(x)(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx=0.}

They are normalized by

∫ − 1 1 [ C n ( α ) ( x ) ] 2 ( 1 − x 2 ) α − 1 2 d x = π 2 1 − 2 α Γ ( n + 2 α ) n ! ( n + α ) [ Γ ( α ) ] 2 . {\displaystyle \int _{-1}^{1}\left[C_{n}^{(\alpha )}(x)\right]^{2}(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx={\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )[\Gamma (\alpha )]^{2}}}.}

Applications

The Gegenbauer polynomials appear naturally as extensions of Legendre polynomials in the context of potential theory and harmonic analysis. The Newtonian potential in Rn has the expansion, valid with α = (n − 2)/2,

1 | x − y | n − 2 = ∑ k = 0 ∞ | x | k | y | k + n − 2 C k ( α ) ( x ⋅ y | x | | y | ) . {\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {y} |^{n-2}}}=\sum _{k=0}^{\infty }{\frac {|\mathbf {x} |^{k}}{|\mathbf {y} |^{k+n-2}}}C_{k}^{(\alpha )}({\frac {\mathbf {x} \cdot \mathbf {y} }{|\mathbf {x} ||\mathbf {y} |}}).}

When n = 3, this gives the Legendre polynomial expansion of the gravitational potential. Similar expressions are available for the expansion of the Poisson kernel in a ball (Stein & Weiss 1971).

It follows that the quantities C k ( ( n − 2 ) / 2 ) ( x ⋅ y ) {\displaystyle C_{k}^{((n-2)/2)}(\mathbf {x} \cdot \mathbf {y} )} are spherical harmonics, when regarded as a function of x only. They are, in fact, exactly the zonal spherical harmonics, up to a normalizing constant.

Gegenbauer polynomials also appear in the theory of positive-definite functions.

The Askey–Gasper inequality reads

∑ j = 0 n C j α ( x ) ( 2 α + j − 1 j ) ≥ 0 ( x ≥ − 1 , α ≥ 1 / 4 ) . {\displaystyle \sum _{j=0}^{n}{\frac {C_{j}^{\alpha }(x)}{2\alpha +j-1 \choose j}}\geq 0\qquad (x\geq -1,\,\alpha \geq 1/4).}

In spectral methods for solving differential equations, if a function is expanded in the basis of Chebyshev polynomials and its derivative is represented in a Gegenbauer/ultraspherical basis, then the derivative operator becomes a diagonal matrix, leading to fast banded matrix methods for large problems.³

Other properties

Dirichlet–Mehler-type integral representation:⁴ P n ( α , α ) ( cos ⁡ θ ) P n ( α , α ) ( 1 ) = C n ( α + 1 2 ) ( cos ⁡ θ ) C n ( α + 1 2 ) ( 1 ) = 2 α + 1 2 Γ ( α + 1 ) π 1 2 Γ ( α + 1 2 ) ( sin ⁡ θ ) − 2 α ∫ 0 θ cos ⁡ ( ( n + α + 1 2 ) ϕ ) ( cos ⁡ ϕ − cos ⁡ θ ) − α + 1 2 d ϕ , {\displaystyle {\frac {P_{n}^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_{n}^{(\alpha ,\alpha )}\left(1\right)}}={\frac {C_{n}^{(\alpha +{\frac {1}{2}})}\left(\cos \theta \right)}{C_{n}^{(\alpha +{\frac {1}{2}})}\left(1\right)}}={\frac {2^{\alpha +{\frac {1}{2}}}\Gamma \left(\alpha +1\right)}{{\pi }^{\frac {1}{2}}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}(\sin \theta )^{-2\alpha }\int _{0}^{\theta }{\frac {\cos \left((n+\alpha +{\tfrac {1}{2}})\phi \right)}{(\cos \phi -\cos \theta )^{-\alpha +{\frac {1}{2}}}}}\,\mathrm {d} \phi ,} Laplace-type P n ( α , α ) ( cos ⁡ θ ) P n ( α , α ) ( 1 ) = C n ( α + 1 2 ) ( cos ⁡ θ ) C n ( α + 1 2 ) ( 1 ) = Γ ( α + 1 ) π 1 2 Γ ( α + 1 2 ) ∫ 0 π ( cos ⁡ θ + i sin ⁡ θ cos ⁡ ϕ ) n ( sin ⁡ ϕ ) 2 α   d ϕ {\displaystyle {\begin{aligned}{\frac {P_{n}^{(\alpha ,\alpha )}(\cos \theta )}{P_{n}^{(\alpha ,\alpha )}(1)}}&={\frac {C_{n}^{\left(\alpha +{\frac {1}{2}}\right)}(\cos \theta )}{C_{n}^{\left(\alpha +{\frac {1}{2}}\right)}(1)}}\\&={\frac {\Gamma (\alpha +1)}{\pi ^{\frac {1}{2}}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\pi }(\cos \theta +i\sin \theta \cos \phi )^{n}(\sin \phi )^{2\alpha }\mathrm {~d} \phi \end{aligned}}}

See also

• Rogers polynomials, the q-analogue of Gegenbauer polynomials
• Chebyshev polynomials
• Romanovski polynomials

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 773. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.*Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9.
• Suetin, P.K. (2001) [1994], "Ultraspherical polynomials", Encyclopedia of Mathematics, EMS Press.

Specific

References

1. Arfken, Weber, and Harris (2013) "Mathematical Methods for Physicists", 7th edition; ch. 18.4 ↩

2. Doha, E. H. (1991-01-01). "The coefficients of differentiated expansions and derivatives of ultraspherical polynomials". Computers & Mathematics with Applications. 21 (2): 115–122. doi:10.1016/0898-1221(91)90089-M. ISSN 0898-1221. https://www.sciencedirect.com/science/article/pii/089812219190089M ↩

3. Olver, Sheehan; Townsend, Alex (January 2013). "A Fast and Well-Conditioned Spectral Method". SIAM Review. 55 (3): 462–489. arXiv:1202.1347. doi:10.1137/120865458. eISSN 1095-7200. ISSN 0036-1445. /wiki/ArXiv_(identifier) ↩

4. "DLMF: §18.10 Integral Representations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials". dlmf.nist.gov. Retrieved 2025-03-18. https://dlmf.nist.gov/18.10 ↩