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Positive discrete Wronskian for the Bessel recurrence pair

Statement

Fix real $x > 0$ , let $J_{j} (x)$ be as in Integer-order Bessel series, and let $D_{j} (x)$ be as in Complementary solution for the integer Bessel recurrence. Define $W_{j} = J_{j} D_{j + 1} - J_{j + 1} D_{j}$ . Then

$W_{j} = J_{0} (x)^{2} + J_{1} (x)^{2} > 0 (j \geq 0) .$

In particular, the two recurrence solutions $J$ and $D$ are linearly independent.

Facts & Assumptions

Given: A real $x > 0$ and the two sequences in the statement.

[F1]

Bessel differential equations and order recurrences from the series: $J_{j + 1} = 2 j J_{j} / x - J_{j - 1}$ for $j \geq 1$ .

[F2]

Complementary solution for the integer Bessel recurrence: $D_{j + 1} = 2 j D_{j} / x - D_{j - 1}$ , with $D_{0} = - J_{1}$ and $D_{1} = J_{0}$ .

[F3]

Adjacent initial Bessel values cannot both vanish: $J_{0} (x)$ and $J_{1} (x)$ are not both zero.

Proof

1.1F1F2algebra

Using [F1] and [F2], for $j \geq 1$ , $W_{j} = J_{j} (\frac{2 j}{x} D_{j} - D_{j - 1}) - (\frac{2 j}{x} J_{j} - J_{j - 1}) D_{j} = J_{j - 1} D_{j} - J_{j} D_{j - 1} = W_{j - 1} .$ Thus $W_{j} = W_{0}$ for every $j \geq 0$ .

2.1F2F3step 1.1algebra∎

By [F2], $W_{0} = J_{0} D_{1} - J_{1} D_{0} = J_{0}^{2} + J_{1}^{2}$ , which is strictly positive by [F3]. A linear dependence of $J, D$ would make every $2 \times 2$ determinant $W_{j}$ zero, so they are independent.

Depends on

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

DLMF, §3.6, Linear Difference Equations (standard reference, not scraped)
N. M. Temme, “Numerical aspects of special functions,” Acta Numerica 17 (2008), §§2 and 4 (standard reference, not scraped)