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31: 30.4 Functions of the First Kind
If γ = 0 , Ps n m ( x , 0 ) reduces to the Ferrers function P n m ( x ) : It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for - 1 x 1 . …
32: 30.1 Special Notation
(For other notation see Notation for the Special Functions.) … The main functions treated in this chapter are the eigenvalues λ n m ( γ 2 ) and the spheroidal wave functions Ps n m ( x , γ 2 ) , Qs n m ( x , γ 2 ) , Ps n m ( z , γ 2 ) , Qs n m ( z , γ 2 ) , and S n m ( j ) ( z , γ ) , j = 1 , 2 , 3 , 4 . …Meixner and Schäfke (1954) use ps , qs , Ps , Qs for Ps , Qs , Ps , Qs , respectively.
Other Notations
33: Bibliography V
  • H. Volkmer (2021) Fourier series representation of Ferrers function P .
  • 34: 10.43 Integrals
    10.43.22 0 t μ - 1 e - a t K ν ( t ) d t = { ( 1 2 π ) 1 2 Γ ( μ - ν ) Γ ( μ + ν ) ( 1 - a 2 ) - 1 2 μ + 1 4 P ν - 1 2 - μ + 1 2 ( a ) , - 1 < a < 1 , ( 1 2 π ) 1 2 Γ ( μ - ν ) Γ ( μ + ν ) ( a 2 - 1 ) - 1 2 μ + 1 4 P ν - 1 2 - μ + 1 2 ( a ) , a 0 , a 1 .
    For the Ferrers function P and the associated Legendre function P , see §§14.3(i) and 14.21(i). …
    35: 30.16 Methods of Computation
    30.16.9 Ps n m ( x , γ 2 ) = lim d j = 1 d ( - 1 ) j - p e j , d P n + 2 ( j - p ) m ( x ) .
    36: 18.3 Definitions
    Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions14.7(i)). …
    37: Mathematical Introduction
    Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). …
    38: Bibliography C
  • H. S. Cohl and R. S. Costas-Santos (2020) Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).
  • H. S. Cohl, J. Park, and H. Volkmer (2021) Gauss hypergeometric representations of the Ferrers function of the second kind. SIGMA Symmetry Integrability Geom. Methods Appl. 17, pp. Paper 053, 33.
  • 39: 18.17 Integrals
    18.17.7 ( P n ( x ) ) 2 + 4 π - 2 ( Q n ( x ) ) 2 = 4 π - 2 1 Q n ( x 2 + ( 1 - x 2 ) t ) ( t 2 - 1 ) - 1 2 d t , - 1 < x < 1 .
    For the Ferrers function Q n ( x ) and Legendre function Q n ( x ) see §§14.3(i) and 14.3(ii), with μ = 0 and ν = n . …
    40: 14.34 Software
    A more complete list of available software for computing these functions is found in the Software Index. For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C9).
    §14.34(ii) Legendre Functions: Real Argument and Parameters
    §14.34(iii) Legendre Functions: Complex Argument and/or Parameters
    §14.34(iv) Conical (Mehler) and/or Toroidal Functions