bibkey: nist2026dlmfmodifiedbessel
authors: NIST Digital Library of Mathematical Functions
year: 2026
title: "Modified Bessel Functions: Definitions and Basic Properties; Connection Formulas"
doi: null
url: https://dlmf.nist.gov/10.25
claim: The first-kind modified Bessel function has its Gamma-denominator defining series; the second-kind function has a connection formula at noninteger orders.
strata_touched:
  - D5/S3/Analytic/BesselProductOrderLogConvexity/Result
license: citation-only
triage: anchor

Modified Bessel functions: definitions and connection formulas

Verified locator

https://dlmf.nist.gov/10.25

DLMF §10.25(ii), equation 10.25.2 (https://dlmf.nist.gov/10.25.E2):

I_ν(z) = (½z)^ν Σ_{k=0}^∞ (¼z²)^k / (k! Γ(ν+k+1)).

The page states that the principal branch of I_ν(z)

is analytic in ℂ∖(−∞,0]

DLMF §10.27(ii), equation 10.27.4 (https://dlmf.nist.gov/10.27.E4):

K_ν(z) = ½π (I_{−ν}(z) − I_ν(z)) / sin(νπ)

followed by

When ν is an integer limiting values are taken

Equation 10.27.1 (https://dlmf.nist.gov/10.27.E1):

I_{−n}(z) = I_n(z), n an integer.

Mathematical correspondence

At real $u>0$, the principal-branch power in equation 10.25.2 is the positive-real power. For $k\in \mathbb{N}$, $(u/2{)}^{\nu }(u^2/4{)}^k=(u/2{)}^{2k+\nu }$. This identifies the defining series with the frozen owner’s D5.S3.Analytic.BesselRatioLogConcavity.Result.besselI.

For real noninteger $\nu$, the denominator $\sin (\nu \pi )$ is nonzero, and equation 10.27.4 identifies D5.S3.Analytic.BesselProductOrderLogConvexity.Result.besselK with the standard second-kind function at $u>0$. The formal assertion uses only noninteger orders. It makes no identification of Lean’s totalized quotient at integer orders with the limiting values prescribed by DLMF.

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