Ferrers function

<h2 id="definitions">Definitions</h2>
<p>Define 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 the order, and the 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 degree are real, and assume 
  
    
      
        x
        ∈
        (
        −
        1
        ,
        +
        1
        )
      
    
    {\displaystyle x\in (-1,+1)}
  
.
</p>
Ferrers function of the first kind

P
          
            v
          
          
            μ
          
        
        (
        x
        )
        =
        
          
            (
            
              
                
                  1
                  +
                  x
                
                
                  1
                  −
                  x
                
              
            
            )
          
          
            μ
            
              /
            
            2
          
        
        ⋅

2
                
              
              
                F
                
                  1
                
              
              (
              v
              +
              1
              ,
              −
              v
              ;
              1
              −
              μ
              ;
              1
              
                /
              
              2
              −
              x
              
                /
              
              2
              )
            
            
              Γ
              (
              1
              −
              μ
              )
            
          
        
      
    
    {\displaystyle P_{v}^{\mu }(x)=\left({\frac {1+x}{1-x}}\right)^{\mu /2}\cdot {\frac {{}_{2}F_{1}(v+1,-v;1-\mu ;1/2-x/2)}{\Gamma (1-\mu )}}}

Ferrers function of the second kind

Q
          
            v
          
          
            μ
          
        
        (
        x
        )
        =
        
          
            π
            
              2
              sin
              ⁡
              (
              μ
              π
              )
            
          
        
        
          (
          
            cos
            ⁡
            (
            μ
            π
            )
            
              
                (
                
                  
                    
                      1
                      +
                      x
                    
                    
                      1
                      −
                      x
                    
                  
                
                )
              
              
                
                  μ
                  2

2
                    
                  
                  
                    F
                    
                      1
                    
                  
                  
                    (
                    
                      v
                      +
                      1
                      ,
                      −
                      v
                      ;
                      1
                      −
                      μ
                      ;
                      
                        
                          
                            1
                            −
                            x
                          
                          2
                        
                      
                    
                    )
                  
                
                
                  Γ
                  (
                  1
                  −
                  μ
                  )
                
              
            
            −
            
              
                
                  Γ
                  (
                  ν
                  +
                  μ
                  +
                  1
                  )
                
                
                  Γ
                  (
                  ν
                  −
                  μ
                  +
                  1
                  )
                
              
            
            
              
                (
                
                  
                    
                      1
                      −
                      x
                    
                    
                      1
                      +
                      x
                    
                  
                
                )
              
              
                
                  μ
                  2

2
                    
                  
                  
                    F
                    
                      1
                    
                  
                  
                    (
                    
                      v
                      +
                      1
                      ,
                      −
                      v
                      ;
                      1
                      +
                      μ
                      ;
                      
                        
                          
                            1
                            −
                            x
                          
                          2
                        
                      
                    
                    )
                  
                
                
                  Γ
                  (
                  1
                  +
                  μ
                  )
                
              
            
          
          )
        
      
    
    {\displaystyle Q_{v}^{\mu }(x)={\frac {\pi }{2\sin(\mu \pi )}}\left(\cos(\mu \pi )\left({\frac {1+x}{1-x}}\right)^{\frac {\mu }{2}}\,{\frac {{}_{2}F_{1}\left(v+1,-v;1-\mu ;{\frac {1-x}{2}}\right)}{\Gamma (1-\mu )}}-{\frac {\Gamma (\nu +\mu +1)}{\Gamma (\nu -\mu +1)}}\left({\frac {1-x}{1+x}}\right)^{\frac {\mu }{2}}\,{\frac {{}_{2}F_{1}\left(v+1,-v;1+\mu ;{\frac {1-x}{2}}\right)}{\Gamma (1+\mu )}}\right)}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Legendre_function/sDo3ra2G">Legendre function</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), "Ferrers Function", NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. <a href="978-0-521-19225-5" target="_blank">978-0-521-19225-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"DLMF: §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions". dlmf.nist.gov. Retrieved 2025-03-17. <a href="https://dlmf.nist.gov/14.3" target="_blank">https://dlmf.nist.gov/14.3</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Ferrers, Norman Macleod. An elementary treatise on spherical harmonics and subjects connected with them. Macmillan and Company, 1877. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Ferrers function open-in-new

Ferrers function