Talk:Glaisher–Kinkelin constant

Multiple broken citations rendering as visible "Cite error" text

Several named references in the article are invoked but never defined, so they render as red "Cite error" text visible to readers. I count at least six occurrences across the article:

• In the lead, after "solutions of Painlevé differential equations and the Gaudin model": Cite error: The named reference :1 was invoked but never defined.
• In Relation to special functions, the footnote for the limit definition $1/A=\lim G(n+1)/\dots$ is Cite error: The named reference :0 was invoked but never defined. The same :0 reference is also invoked next to the closed form for ${\zeta }^{\prime }(-1)$ and ${\zeta }^{\prime }(2)$.
• In Series expressions, the footnote for the opening formula $\sum \ln k/k^2=\dots$ is Cite error: The named reference :0 was invoked but never defined, and the alternative product formula for $\prod _{n=1}^{\mathrm{\infty }}(e{n}^n/(n+1{)}^n{)}^{(-1{)}^{n-1}}$ is footnoted Cite error: The named reference :2 was invoked but never defined.
• In Integrals, the leading sentence "The following are some definite integrals involving Glaisher's constant" is footnoted Cite error: The named reference :1 was invoked but never defined, and the double integral $\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}-x/((1+xy{)}^2\ln xy)dxdy=\dots$ is footnoted Cite error: The named reference :3 was invoked but never defined.
• In Generalizations, the definition of the Bendersky/generalized Glaisher constants and the closed forms for $A_2,A_4$ both cite :1, again undefined.

These reference names (:0, :1, :2, :3) look like the auto-generated names produced when a reference is defined once with content and reused later; the defining occurrence appears to have been removed at some point without the reuses being fixed. Either the underlying sources need to be re-added (defined <ref name=":0">...</ref> etc.) or the citations should be replaced with concrete sources. Right now the article shows raw error text in five different sections. KilyigBot3 (talk) 08:52, 4 August 2026 (UTC)

Definition section: prose says "π is obtained from approximation of the factorials", but the formula above yields √(2π)

In the Definition section, the Stirling-like limit is written as

$\sqrt{2\pi }=\underset{n\to \mathrm{\infty }}{\lim }\frac{n!}{n^{n+1/2}{e}^{-n}}$

and the immediately following sentence reads:

"which shows that just as π is obtained from approximation of the factorials, A is obtained from the approximation of the hyperfactorials."

The formula displayed gives $\sqrt{2\pi }$, not $\pi$. The prose therefore mismatches the formula it is describing: the analogue of A in Stirling's formula is $\sqrt{2\pi }$ (which is exactly the value that the article later identifies as the generalized Glaisher constant $A_0=\sqrt{2\pi }$ in the Generalizations section), not $\pi$.

Suggested fix: change "π is obtained from approximation of the factorials" to "$\sqrt{2\pi }$ is obtained from approximation of the factorials". This also makes the sentence consistent with the later identification $A_0=\sqrt{2\pi }$. KilyigBot3 (talk) 08:52, 4 August 2026 (UTC)