Irrationality proofs à la Hermite

@article{Zhou2011IrrationalityP,
  title={Irrationality proofs {\`a} la Hermite},
  author={Li Zhou},
  journal={The Mathematical Gazette},
  year={2011},
  volume={95},
  pages={407 - 413}
}
  • Li Zhou
  • Published 10 November 2009
  • Mathematics
  • The Mathematical Gazette
In [1] Niven used the integral to give a well-known proof of the irrationality of π. Recently Zhou and Markov [2] used a recurrence relation satisfied by this integral to present an alternative proof which may be more direct than Niven's. Niven did not cite any references in [1] and thus the origin or Hn seems rather mysterious and ingenious. However if we heed Abel's advice to ‘study the masters’, we find that Hn emerged much more naturally from the great works of Lambert [3] and Hermite [4]. 
1 Citations
ON “DISCOVERING AND PROVING THAT π IS IRRATIONAL”
We discuss the logical fallacies in an article appeared in The American Mathematical Monthly [6], and present the historical origin and motivation of the simple proofs of the irrationality of π. 1. A

References

SHOWING 1-10 OF 18 REFERENCES
Recurrent Proofs of the Irrationality of Certain Trigonometric Values
We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to
A Course of Modern Analysis
TLDR
The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.
Extrait d'une lettre de Monsieur Ch. Hermite à Monsieur Paul Gordan.
En attendant, c'est des fractions continues algebriques que je prends la liberte de vous entretenir, ou plutot d'ime extension de cette theorie, ayant cherche le Systeme des polyn mes entiers en x,
A Simple Proof that π is Irrational
Let π = a/b, the quotient of positive integers. We define the polynomials $$ \begin{array}{*{20}{c}} {f(x) = \frac{{{x^n}{{(a - bx)}^n}}}{{n!}},} \\ {F(x) = f(x) - {f^{(2)}}(x) + {f^4}(x) - + {{(
Mémoires sur quelques propriétés remarquables des quantités transcendantes
  • circulaires et logarithmiques, Mém. de l’Acad. R. des Sci. de Berlin 17
  • 1768
Irresistible integrals
  • Cambridge Univ. Press
  • 2004
A course of modem analysis
  • 2002
A course of modem analysis (4th edn.)
  • 2002
Mémoire Sur Quelques Propriétés Remarquables des Quantités Transcendentes Circulaires et Logarithmiques
Demontrer que le diametre da cercle n’eft point a fa circonference comme un nombre entier a un nombre entier, c’eft la unechofe, dont les geometres ne feront gueres farpris. On connoit les nombres de
...
1
2
...