A Differential Geometric Proof Of The Complete Formula For Legendre Polynomials

The study of Legendre polynomials can be traced back to the last century.Up to now,the determination of Legendre polynomials'coefficients is proved by the method of harmonic analysis,but there is no proof of differential geometry.1.In this paper,a new unified differential geometry proof for the determination of all coefficients of Legendre polynomials is given,As a special case,the complete formula of Chebyshev polynomials is obtained.Although our result is equal to the classical result,the expression is different from the classical one.The characteristic root of spherical Laplace naturally appears in the result.2.The coefficients of Legendre polynomials expressed by Bernstein polynomials are determined,thus obtain a second expression of Legendre polynomials.3.Some combinatorial identities are obtained.The research methods are as follows:In N(d,n)=Cd+n-1d-1-Cd+n-3d-1 dimen-sional spherical harmonic function space Hnd,choose any set of orthonormal basis Sjn(?).The elements in Hnd are the restriction to Sd-1 of homogeneous harmonic polynomi-als in Rnd.It can be proved that there exists a polynomial Q(t)of degree n,such that?jSjn(?)Sjn(?)=Q(t),t=?·?.Let Pnd(t)=-?d-1/NQ(t),then Pnd(t)is called Leg-endre polynomial.Because Sd-1 is a submanifold of Rd,We also obtain the classical relations between Laplacian operators of functions in sd-1 and Rd by using the moving frame method in differential geometry.It is concluded that all the elements in Hnd are eigen-functions of spherical Laplacian.Note that the independent variable t in Legendre polynomials is originally the inner product of any two points ? and ? on the sphere.So,Legendre polynomials are functions of ? and ?.It is not only a spherical harmonic of order n about ?,but also a spherical harmonic of ?.So when Legendre polynomials are rewritten as Pnd(?·?),it is the eigen-function of the spherical Laplacian of ? or ?,(?)By using the result of Laplacian and gradient of coordinate function on sphere due to Wu Faen in 2005,we can calculate the Laplacian of Pnd(?·?).By comparing the coefficients,the recurrence formula between the coefficients is obtained.In the expression of Leg-endre polynomials,let the dimension of Euclidean space be d=2 to obtain Chebyshev polynomials.Wang Kunyang and Li Luoqing mentioned in their works that Legendre polynomials can be expressed as polynomials of Bernstein polynomials,but no specific expression is given.Note that Legendre polynomials can be expressed as polynomials of Bernstein polynomials means,when n is even,after making a variable replacement,the expression can be expressed by a set of Bernstein polynomials;when n is odd,it can be expressed by the product of independent variable t and similar expression when n is even.In this paper,by using F.Gauss representation of Legendre polynomials,the bino-mial theorem and the volume formula of any dimensional unit sphere,the coefficients of Legendre polynomials expressed by Bernstein polynomials are obtained.By comparing the coefficients of the two expressions of Legendre polynomials,we get some combina-torial identities.