Add tests for the Jacobi ladder operators

src/ClassicalOrthogonalPolynomials.jl

@@ -255,6 +255,10 @@
     C*jacobimatrix(C)
 end
 
+function layout_broadcasted(::Tuple{ScalarLayout,AbstractOPLayout}, ::typeof(*), c, C)
+    C * c
+end
+
 
 # function broadcasted(::LazyQuasiArrayStyle{2}, ::typeof(*), a::BroadcastQuasiVector, C::OrthogonalPolynomial)
 #     axes(a,1) == axes(C,1) || throw(DimensionMismatch())

src/classical/jacobi.jl

@@ -485,6 +485,10 @@
 broadcastbasis(::typeof(+), A::Jacobi, B::Weighted{<:Any,<:Jacobi{<:Any,<:Integer}}) = A
 broadcastbasis(::typeof(+), A::Weighted{<:Any,<:Jacobi{<:Any,<:Integer}}, B::Jacobi) = B
 
+broadcasted(::LazyQuasiArrayStyle{2}, ::typeof(*), w::AbstractJacobiWeight, v::WeightedJacobi) =
+    (w .* v.args[1]) .* v.args[2]
+
+
 function \(w_A::WeightedJacobi, w_B::WeightedJacobi)
     wA,A = w_A.args
     wB,B = w_B.args

test/test_jacobi.jl

@@ -552,4 +552,30 @@ import ClassicalOrthogonalPolynomials: recurrencecoefficients, basis, MulQuasiMa
         @test expand(W, x -> (1-x^2)*exp(x))[0.1] ≈ (1-0.1^2)*exp(0.1)
         @test grid(W, 5) == grid(W.P, 5)
     end
+
+    @testset "ladder operators" begin
+        a,b = 0.1,0.2
+        P = Jacobi(a,b)
+        A₊,A₋ = Jacobi(a+1,b),Jacobi(a-1,b)
+        B₊,B₋ = Jacobi(a,b+1),Jacobi(a,b-1)
+        C₊,C₋ = Jacobi(a+1,b+1),Jacobi(a-1,b-1)
+        D₊,D₋ = Jacobi(a+1,b-1), Jacobi(a-1,b+1)
+        t,n = 0.3,5
+        x = axes(P,1)
+        D = Derivative(P)
+
+        @test (D*P)[t,n+1] ≈ (n+a+b+1)/2 * C₊[t,n] # L₁
+        @test (((a+b+n+1)*I + (1 .+ x) .* D)*P)[t,n+1] ≈ (JacobiWeight(0,-(a+b+n)) .* D * (JacobiWeight(0,a+b+n+1) .* P))[t,n+1] ≈ (n+a+b+1) * A₊[t,n+1] # L₂
+        @test (((a+b+n+1)*I - (1 .- x) .* D)*P)[t,n+1] ≈ -(JacobiWeight(-(a+b+n),0) .* D * (JacobiWeight(a+b+n+1,0) .* P))[t,n+1] ≈ (n+a+b+1) * B₊[t,n+1] # L₃
+        @test ((1 .+ t)*a - (1 .- t)*(b+n+1))*P[t,n+1] - (((1 .- x.^2) .* D)*P)[t,n+1] ≈ -(JacobiWeight(1-a,-(b+n)) .* D * (JacobiWeight(a,b+n+1) .* P))[t,n+1] ≈ 2*(n+1) * A₋[t,n+2] # L₄
+        @test ((1 .+ t)*(a+n+1) - (1 .- t)*b)*P[t,n+1] - (((1 .- x.^2) .* D)*P)[t,n+1] ≈ -(JacobiWeight(-(a+n),1-b) .* D * (JacobiWeight(a+n+1,b) .* P))[t,n+1] ≈ 2*(n+1) * B₋[t,n+2] # L₅
+        @test ((b*I + (1 .+ x) .* D)*P)[t,n+1] ≈ (JacobiWeight(0,1-b) .* D * (JacobiWeight(0,b) .* P))[t,n+1] ≈ (n+b) * D₊[t,n+1] # L₆
+
+        @test ((1 .+ t)*a - (1 .- t)*b)*P[t,n+1] - (((1 .- x.^2) .* D)*P)[t,n+1] ≈ -(JacobiWeight(1-a,1-b) .* D * (JacobiWeight(a,b) .* P))[t,n+1] ≈ 2*(n+1) * C₋[t,n+2] # L₁'
+        @test (2a + (1 .- t)*n)*P[t,n+1] - (((1 .- x.^2) .* D)*P)[t,n+1] ≈ -(JacobiWeight(1-a,1+a+n) .* D * (JacobiWeight(a,-a-n) .* P))[t,n+1] ≈ 2*(n+a) * A₋[t,n+1] # L₂'
+        @test (2b + (1 .+ t)*n)*P[t,n+1] + (((1 .- x.^2) .* D)*P)[t,n+1] ≈ (JacobiWeight(1+b+n,1-b) .* D * (JacobiWeight(-b-n,b) .* P))[t,n+1] ≈ 2*(n+b) * B₋[t,n+1] # L₃'
+        @test ((-n*I + (1 .+ x) .* D)*P)[t,n+1] ≈ (JacobiWeight(0,n+1) .* D * (JacobiWeight(0,-n) .* P))[t,n+1] ≈ (n+b) * A₊[t,n] # L₄'
+        @test ((n*I + (1 .- x) .* D)*P)[t,n+1] ≈ (JacobiWeight(n+1,0) .* D * (JacobiWeight(-n,0) .* P))[t,n+1] ≈ (n+a) * B₊[t,n] # L₅'
+        @test ((a*I - (1 .- x) .* D)*P)[t,n+1] ≈ -(JacobiWeight(1-a,0) .* D * (JacobiWeight(a,0) .* P))[t,n+1] ≈ (n+a) * D₋[t,n+1] # L₆'
+    end
 end

test/test_ultraspherical.jl

@@ -204,4 +204,35 @@ using ClassicalOrthogonalPolynomials: grammatrix
         @test (C \ diff(U,1))[1:10,1:10] == (C \ diff(U))[1:10,1:10]
         @test (C³ \ diff(U,2))[1:10,1:10] == (C³ \ diff(diff(U)))[1:10,1:10]
     end
+
+    @testset "ladder" begin
+        λ = 3/4
+        P = Jacobi(λ-1/2,λ-1/2)
+        W = Jacobi(λ-1-1/2,λ-1-1/2)
+        Q = Jacobi(λ+1-1/2,λ+1-1/2)
+        # a+ b = 2λ-1
+        D = Derivative(P)
+        x,n = 0.3,5
+        # L₁
+        @test (D*P)[t,n+1] ≈ (n+2λ)/2 * Q[t,n] # L₁
+        # L₃L₂
+        @test -(JacobiWeight(-(2λ+n),0) .* (D * (JacobiWeight(2λ+n+1,-(2λ+n-1)) .* D * (JacobiWeight(0,2λ+n) .* P))))[x,n+1] ≈ 
+        (n + 2λ)*(1 + n + 2λ) * P[x,n+1] + 2x*(1+n+2λ)* diff(P)[x,n+1] + (x^2-1) * diff(P,2)[x,n+1] ≈ (n+2λ)*(n+2λ+1)*Q[x,n+1]
+        # L₄'L₃ == L₅'L₂
+        @test -(JacobiWeight(0,n+1) .* (D * (JacobiWeight(-(2λ+n-1),-n) .* D * (JacobiWeight(2λ+n,0) .* P))))[x,n+1] ≈ 
+        (JacobiWeight(n+1,0) .* (D * (JacobiWeight(-n,-(2λ+n-1)) .* D * (JacobiWeight(0,2λ+n) .* P))))[x,n+1] ≈ 
+        -n * (n + 2λ) * P[x,n+1] + (1 + 2n + x + 2λ + 2x*λ)* diff(P)[x,n+1] + (x^2-1) * diff(P,2)[x,n+1] ≈
+        n * (n + 2λ) * P[x,n+1] + (1 + 2n - x + 2λ - 2x*λ)* diff(P)[x,n+1] + (1-x^2) * diff(P,2)[x,n+1] ≈ (n+2λ)*(n+λ+1/2)*Q[x,n]
+        # L₅'L₄' == L₄'L₅'
+        @test (JacobiWeight(n,0) .* (D * (JacobiWeight(1-n,n+1) .* D * (JacobiWeight(0,-n) .* P))))[x,n+1] ≈ 
+            (JacobiWeight(0,n) .* (D * (JacobiWeight(n+1,1-n) .* D * (JacobiWeight(-n,0) .* P))))[x,n+1] ≈ 
+            -n * (n - 1) * P[x,n+1] + 2x * (n-1)* diff(P)[x,n+1] - (x^2-1) * diff(P,2)[x,n+1] ≈
+            (n+λ-1/2)^2*Q[x,n-1]
+
+        P = Ultraspherical(λ)
+        Q = Ultraspherical(λ+1)
+        # L₃L₂
+        @test (n + 2λ)*(1 + n + 2λ) * P[t,n+1] + 2t*(1+n+2λ)* diff(P)[t,n+1] + (t^2-1) * diff(P,2)[t,n+1] ≈ 2λ*(1 + 2n + 2λ)Q[t,n+1]
+
+    end
 end