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1: 18.1 Notation
x -Differences
Forward differences: … Backward differences: … Central differences in imaginary direction: … In Koekoek et al. (2010) δ x denotes the operator i δ x .
2: 18.2 General Orthogonal Polynomials
§18.2(ii) x -Difference Operators
If the orthogonality discrete set X is { 0 , 1 , , N } or { 0 , 1 , 2 , } , then the role of the differentiation operator d / d x in the case of classical OP’s (§18.3) is played by Δ x , the forward-difference operator, or by x , the backward-difference operator; compare §18.1(i). … If the orthogonality interval is ( - , ) or ( 0 , ) , then the role of d / d x can be played by δ x , the central-difference operator in the imaginary direction (§18.1(i)). …
3: 18.20 Hahn Class: Explicit Representations
For comments on the use of the forward-difference operator Δ x , the backward-difference operator x , and the central-difference operator δ x , see §18.2(ii). …
4: 18.26 Wilson Class: Continued
For comments on the use of the forward-difference operator Δ x , the backward-difference operator x , and the central-difference operator δ x , see §18.2(ii). …
18.26.16 Δ y ( R n ( y ( y + γ + δ + 1 ) ; α , β , γ , δ ) ) Δ y ( y ( y + γ + δ + 1 ) ) = n ( n + α + β + 1 ) ( α + 1 ) ( β + δ + 1 ) ( γ + 1 ) R n - 1 ( y ( y + γ + δ + 2 ) ; α + 1 , β + 1 , γ + 1 , δ ) .
18.26.17 Δ y ( R n ( y ( y + γ + δ + 1 ) ; γ , δ , N ) ) Δ y ( y ( y + γ + δ + 1 ) ) = - n ( γ + 1 ) N R n - 1 ( y ( y + γ + δ + 2 ) ; γ + 1 , δ , N - 1 ) .
5: 3.9 Acceleration of Convergence
3.9.2 S = k = 0 ( - 1 ) k 2 - k - 1 Δ k a 0 ,
Here Δ is the forward difference operator:
3.9.3 Δ k a 0 = Δ k - 1 a 1 - Δ k - 1 a 0 , k = 1 , 2 , .
3.9.4 Δ k a 0 = m = 0 k ( - 1 ) m ( k m ) a k - m .
6: Mathematical Introduction

complex plane (excluding infinity).

Δ (or Δ x )

forward difference operator: Δ f ( x ) = f ( x + 1 ) - f ( x ) .

(or x )

backward difference operator: f ( x ) = f ( x ) - f ( x - 1 ) . (See also del operator in the Notations section.)

7: 3.6 Linear Difference Equations
3.6.2 a n Δ 2 w n - 1 + ( 2 a n - b n ) Δ w n - 1 + ( a n - b n + c n ) w n - 1 = d n ,
where Δ w n - 1 = w n - w n - 1 , Δ 2 w n - 1 = Δ w n - Δ w n - 1 , and n . …
8: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.19 Δ x Q n ( x ; α , β , N ) = - n ( n + α + β + 1 ) ( α + 1 ) N Q n - 1 ( x ; α + 1 , β + 1 , N - 1 ) ,
18.22.21 Δ x K n ( x ; p , N ) = - n p N K n - 1 ( x ; p , N - 1 ) ,
18.22.23 Δ x M n ( x ; β , c ) = - n ( 1 - c ) β c M n - 1 ( x ; β + 1 , c ) ,
18.22.25 Δ x C n ( x ; a ) = - n a C n - 1 ( x ; a ) ,
9: 18.25 Wilson Class: Definitions
For the Wilson class OP’s p n ( x ) with x = λ ( y ) : if the y -orthogonality set is { 0 , 1 , , N } , then the role of the differentiation operator d / d x in the Jacobi, Laguerre, and Hermite cases is played by the operator Δ y followed by division by Δ y ( λ ( y ) ) , or by the operator y followed by division by y ( λ ( y ) ) . Alternatively if the y -orthogonality interval is ( 0 , ) , then the role of d / d x is played by the operator δ y followed by division by δ y ( λ ( y ) ) . …
10: 2.9 Difference Equations
2.9.2 Δ 2 w ( n ) + ( 2 + f ( n ) ) Δ w ( n ) + ( 1 + f ( n ) + g ( n ) ) w ( n ) = 0 , n = 0 , 1 , 2 , ,
in which Δ is the forward difference operator3.6(i)). …