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In 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirlings approximation of $n!$. He gave a good formula which is asymptotic to $n!$. Since then hundreds of papers have given alternative proofs of his result and improved upon it, including notably by Burside, Gosper, and Mortici. However Srinivasa Ramanujan gave a remarkably better asymptotic formula. Hirschhorn and Villarino gave a nice proof of Ramanujans result and an error estimate for the approximation. In recent years there have been several improvements of Stirlings formula including by Nemes, Windschitl, and Chen. Here it is shown (i) how all these asymptotic results can be easily verified; (ii) how Hirschhorn and Villarinos argument allows a tweaking of Ramanujans result to give a better approximation; (iii) that a new asymptotic formula can be obtained by further tweaking of Ramanujans result; (iv) that Chens asymptotic formula is better than the others mentioned here, and the new asymptotic formula is comparable with Chens.
Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are replaced with
Using Ramanujans identities and the Weierstrass-Enneper representation of minimal surfaces and the analogue for Born-Infeld solitons, we derive further non-trivial identities.
By virtue of Baileys well-known bilateral 6psi_6 summation formula and Watsons transformation formula,we extend the four-variable generalization of Ramanujans reciprocity theorem due to Andrews to a five-variable one. Some relevant new q-series ident
We consider natural variants of Lehmers unresolved conjecture that Ramanujans tau-function never vanishes. Namely, for $n>1$ we prove that $$tau(n) ot in {pm 1, pm 3, pm 5, pm 7, pm 691}.$$ This result is an example of general theorems for newforms w
We study the relation between Hecke groups and the modular equations in Ramanujans theories of signature 2, 3 and 4. The solution $(alpha,beta)$ to the generalized modular equation satisfies a polynomial equation $P(alpha,beta)=0$ and we determine th