Express J₄(x) in terms of J₁ and J₀

[Printed coefficient: 3/x of J₁. Working with the recurrence relation gives 8/x, so the 3/x appears to be a misprint; the identity is proved below with 8/x, i.e. J₄ = (48/x³ − 8/x)J₁ + (1 − 24/x²)J₀. Checked numerically at x = 2.3, where both sides equal 0.05560, while the 3/x version gives 1.229.]

Use the recurrence relation Jₚ₊₁ = (2p/x)Jₚ − Jₚ₋₁.

Step by step

J₂ = (2/x)J₁ − J₀                                  (p = 1)
J₃ = (4/x)J₂ − J₁                                  (p = 2)
   = (4/x)[(2/x)J₁ − J₀] − J₁
   = (8/x² − 1)J₁ − (4/x)J₀
J₄ = (6/x)J₃ − J₂                                  (p = 3)
   = (6/x)[(8/x² − 1)J₁ − (4/x)J₀] − [(2/x)J₁ − J₀]
   = (48/x³ − 6/x − 2/x)J₁ + (1 − 24/x²)J₀

Result: J₄(x) = (48/x³ − 8/x) J₁(x) + (1 − 24/x²) J₀(x).

The J₀ coefficient (1 − 24/x²) matches the printed form; only the J₁ coefficient differs (8/x, not 3/x).