§ 2 Orthogonal polynomials

 

1.        Legendre polynomial

 

[ Generating function of Legendre polynomial ] is expanded by function press :

     

to define the sequence of Legendre polynomials

    The function is called a generating or generating function .

    [ Expression of Legendre polynomial ]

          

                            

                                     

                        

                            

           

           

        ･･･････････････････････････････････     

                       (Merfeit expression)

   

   

   

[ Legendre differential equations ]

   

[ Orthogonality of Legendre Polynomials ]

     

[ Inequalities and Special Values ]

                                            

            

                  

   

            

[ Recursion formula and derivative formula ]

          (recursion relationship)

   

   

   

   

 

2.        Chebyshev polynomials of the first kind

 

[ Generating function of Chebyshev polynomials of the first kind ] is expanded   by the generating function :

          

to define a sequence of Chebyshev polynomials of the first kind .

    [ Expressions for Chebyshev polynomials of the first kind ]

       

                   

                   

       

                                     

                                     

                             

                           

                     

                

        ････････････････････････････････････    

    [ Chebyshev differential equations of the first kind ]

       

    [ Orthogonality of Chebyshev Polynomials of the First Kind ]

       

    [ Inequalities and Special Values ]

                                        

             

    [ Recursion formula and derivative formula ]

                         (recursive formula)

       

                             

 

3.        Chebyshev polynomials of the second kind

 

[ Generating function of Chebyshev polynomials of the second kind ] is expanded by the generating function :

           

to define a sequence of Chebyshev polynomials of the second kind .

[ Expressions for Chebyshev polynomials of the second kind ]

   

            

            

   

   

   

   

   

   

   

    ……………………

[ Chebyshev differential equations of the second kind ]

   

[ Orthogonality of Chebyshev Polynomials of the Second Kind ]

   

[ Inequalities and Special Values ]

                        

       

[ Recursive formula and related formulas ]

           (recursive formula)

   

   

   

   

 

4.        Laguerre polynomials

 

1.        General Laguerre polynomials

[ Generic function of a general Laguerre polynomial ] is expanded by the generating function :

              

to define a general sequence of Laguerre polynomials .

[ Expression of general Laguerre polynomial ]

   

         

         

      

   

   

where is the Kummer function, which is a first-order Bessel function. very

                    

[ General Laguerre Differential Equations ]

   

[ Orthogonality of General Laguerre Polynomials ]

   

[ Inequalities and Special Values ]

   

   

[ Recursive formula and related formulas ]

        (recursive formula)

   

   

               

   

   

   

   

where is the Hermitian polynomial.

2.          Laguerre polynomials

In general Laguerre polynomials, then , define

                                       

is the Laguerre polynomial . Its corresponding formula is

              (generating function expansion)

             

             

       

       

       

       

       

        ････････････････････････････････

                     (Laguerre differential equations)

                              (orthogonality)

              

                       (recursive formula)

 

5.        Hermitian polynomials

 

[ Generating function of Hermitian polynomial ] is expanded by the generating function :

                    

to define a sequence of Hermitian polynomials .

[ Expression of Hermitian polynomial ]

   

   

        ･･････････

   

   

where is the Kummer function .

       

[ Asymptotic expressions for Hermitian polynomials ]

           

       

       

       

             

[ Hermitian differential equations ]

                 

   [ Orthogonality of Hermitian Polynomials ]

                 

   [ Inequalities and Special Values ]

       

          

              

       

                  

    [ Recursive formula and related formulas ]

                        (recursive formula)

       

             

    [ weighted Hermitian polynomial ]   is the Hermitian polynomial of the weight function,

Its expression is

       

    relationship with

                                 

 

Six,        Jacobi polynomial

 

[ Generating function of Jacobian polynomial ] is expanded by the generating function (where ):

       

to define a sequence of Jacobi polynomials .

[ Expression for Jacobian polynomial ]

   

           

   

             

where F is the hypergeometric function.

    [ Jacobi Differential Equations ]

            

    [ Orthogonality of Jacobian Polynomials ]

       

                           

    [ Inequalities and Special Values ]

       

where is one of the two maxima points closest to the point .

                  

                           

[ Recursive formula and related formulas ]

   

      

            

                                                                                 (recursive formula)

   

 

7.        Geigenberger polynomial

 

[ Generating function of Geigenberger polynomial ]   Expansion by the generating function

        

to define the sequence of Geigenberger polynomials, also known as special spherical polynomials .

[ Expression of Geigenberger polynomial ]

    

          

          

       

               

                                          

where is the hypergeometric function .

       

        ······························································································ ･･･

[ Gegenberg differential equations ]

   

[ Orthogonality of Geigenberg Polynomials ]

   

[ Inequalities and Special Values ]

                                  

               and not an integer)

   

                                   ( not an integer, and

   

   

   

[ Recursive formula and related formulas ]

        (recursive formula)