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Euler–Poisson–Darboux equation

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1: 24.1 Special Notation
Euler Numbers and Polynomials
Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
2: 30.2 Differential Equations
§30.2(i) Spheroidal Differential Equation
The equation contains three real parameters λ , γ 2 , and μ . … With ζ = γ z Equation (30.2.1) changes to … If γ = 0 , Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). …If γ = 0 , Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).
3: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
§15.10(i) Fundamental Solutions
15.10.1 z ( 1 z ) d 2 w d z 2 + ( c ( a + b + 1 ) z ) d w d z a b w = 0 .
This is the hypergeometric differential equation. …
4: 31.2 Differential Equations
§31.2(i) Heun’s Equation
This equation has regular singularities at 0 , 1 , a , , with corresponding exponents { 0 , 1 γ } , { 0 , 1 δ } , { 0 , 1 ϵ } , { α , β } , respectively (§2.7(i)). … The parameters play different roles: a is the singularity parameter; α , β , γ , δ , ϵ are exponent parameters; q is the accessory parameter. … w ( z ) = z 1 γ w 1 ( z ) satisfies (31.2.1) if w 1 is a solution of (31.2.1) with transformed parameters q 1 = q + ( a δ + ϵ ) ( 1 γ ) ; α 1 = α + 1 γ , β 1 = β + 1 γ , γ 1 = 2 γ . …
5: 29.2 Differential Equations
§29.2 Differential Equations
§29.2(i) Lamé’s Equation
§29.2(ii) Other Forms
Equation (29.2.10) is a special case of Heun’s equation (31.2.1).
6: 32.2 Differential Equations
The six Painlevé equations P I P VI  are as follows: … with α , β , γ , and δ arbitrary constants. … For arbitrary values of the parameters α , β , γ , and δ , the general solutions of P I P VI  are transcendental, that is, they cannot be expressed in closed-form elementary functions. … If γ δ 0 in P III , then set γ = 1 and δ = 1 , without loss of generality, by rescaling w and z if necessary. …Lastly, if δ = 0 and β γ 0 , then set β = 1 and γ = 1 , without loss of generality. …
7: 28.2 Definitions and Basic Properties
§28.2(i) Mathieu’s Equation
This is the characteristic equation of Mathieu’s equation (28.2.1). …
§28.2(iv) Floquet Solutions
8: 28.20 Definitions and Basic Properties
§28.20(i) Modified Mathieu’s Equation
When z is replaced by ± i z , (28.2.1) becomes the modified Mathieu’s equation:
28.20.1 w ′′ ( a 2 q cosh ( 2 z ) ) w = 0 ,
28.20.2 ( ζ 2 1 ) w ′′ + ζ w + ( 4 q ζ 2 2 q a ) w = 0 , ζ = cosh z .
Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ζ 1 / 2 e ± 2 i h ζ as ζ in the respective sectors | ph ( i ζ ) | 3 2 π δ , δ being an arbitrary small positive constant. …
9: 19.18 Derivatives and Differential Equations
and two similar equations obtained by permuting x , y , z in (19.18.10). More concisely, if v = R a ( 𝐛 ; 𝐳 ) , then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation: …and also a system of n ( n 1 ) / 2 EulerPoisson differential equations (of which only n 1 are independent): … The next four differential equations apply to the complete case of R F and R G in the form R a ( 1 2 , 1 2 ; z 1 , z 2 ) (see (19.16.20) and (19.16.23)). The function w = R a ( 1 2 , 1 2 ; x + y , x y ) satisfies an EulerPoissonDarboux equation: …
10: 18.2 General Orthogonal Polynomials
§18.2(v) Christoffel–Darboux Formula
Confluent Form
Poisson kernel
For OP’s p n with h n and orthogonality relation as in (18.2.5) and (18.2.5_5), the Poisson kernel is defined by …Instances where the Poisson kernel is nonnegative are of special interest, see Ismail (2009, Theorem 4.7.12). …