## Partitions without small parts.(English)Zbl 0707.11072

Number theory. Vol. I. Elementary and analytic, Proc. Conf., Budapest/Hung. 1987, Colloq. Math. Soc. János Bolyai 51, 9-33 (1990).
Let $$r(n,m)$$ denote the number of partitions of $$n$$ into parts $$\ge m$$. The present authors show that the asymptotic formula $r(n,m)=p(n)\left(\frac{C}{2\sqrt{n}}\right)^{m-1}(m-1)!\{1+O(m^2/\sqrt{n})\} \tag{*}$ holds uniformly for $$1\le m\le n^{1/4}$$; here $$C=\pi \sqrt{2/3}$$ and $$p(n)=r(n,1)$$ is the number of (unrestricted) partitions of $$n$$. Their proof is based on the representation $$r(n,m)=D^{(m-1)}(p(n))$$ where $$D^{(m-1)}$$ is a difference operator reflecting the relation $\sum_{n\geq 1}r(n,m)x^ n=\prod^{m-1}_{k-1}(1-x^ k)\sum_{n\geq 1}p(n)x^ n.$ Here $$p(n)$$ may be replaced by the Hardy-Ramanujan formula $p(n)=\frac{C^3}{2\pi \sqrt{2}}F'(C^2(n-1/24)) + \text{error}$ where $$F(x)=\exp (\sqrt{x})/\sqrt{x}$$, hence a simple extension of the mean value theorem gives for $$m\le \sqrt{n}$$ $r(n,m)=(m-1)!\frac{C^{2m+1}}{2\pi \sqrt{2}}F^{(m)}(t) + \text{error}$ with $$C^2(n- m^2/2)\le t\le C^2n$$. Evaluation of $$F^{(m)}(t)$$ in terms of Bessel polynomials and some straightforward estimates yield (*) uniformly in $$m\le n^{1/4}$$.
As an application of (*) an asymptotic formula of P. Erdős and M. Szalay [Studies in Pure Math., Memory of P. Turán, 187–212 (1983; Zbl 0523.10029)] for the number of nonpractical partitions of $$n$$ is considerably sharpened. (A partition $$n=n_1+\cdots+n_k$$ of $$n$$ is said to be practical if each $$b\in \{1,\ldots,n\}$$ is representable as a subsum $$b=\sum^k_{i=1}a_i n_i$$ with $$a_i\in \{0,1\}$$.)
[For the entire collection see Zbl 0694.00005.]

### MSC:

 11P82 Analytic theory of partitions

### Citations:

Zbl 0694.00005; Zbl 0523.10029