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Asymptotic formula for the number of solutions of a diophantic system. (English) Zbl 0607.10033

For positive integers \(n\), \(N_1\), \(N_2\) and \(p\), let \(R_{n,p}(N_1,N_2)\) denote the number of solutions of the following diophantic system: \[ (\text{i})\quad 1x_1+2x_2+ \ldots +nx_n=N_1,\quad (\text{ii})\quad x_1+x_2+ \ldots +x_n=N_2,\quad (\text{iii})\quad 0\leq x_ j\leq p\quad (j=1,2,\ldots,n). \] This problem is motivated by an application to Wilcoxon statistics but every subsystem and several special cases of this system have been investigated in additive number theory. For \(R_{N_1,N_1}(N_1,N_2)\) and \(R_{N_1,1}(N_1,N_2)\), see P. Erdős and J. Lehner [Duke Math. J. 8, 335–345 (1941; Zbl 0025.10703)], G. Szekeres [Q. J. Math., Oxf. II. Ser. 2, 85–108 (1951; Zbl 0042.04102); ibid. 4, 96–111 (1953; Zbl 0050.04101)]. As to the subsystem (ii), (iii), we can refer to G. Castelnuovo [Ann. Inst. H. Poincaré 3, 465–490 (1933; Zbl 0007.25301)], A. G. Postnikov [Izv. Akad. Nauk SSSR, Ser. Mat. 20, 751–764 (1956; Zbl 0075.03401)], S. Kh. Sirazhdinov, T. A. Azlarov and T. M. Zuparov [Additive problems with an increasing number of terms (Russian) Tashkent: FAN (1975; Zbl 0418.10023)]. For the subsystem (i), (iii), I. Joó [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 28(1986), 217–227 (1985; Zbl 0607.10032), see the preceding review] proved a uniform estimation.
The main result of the paper under review is Theorem 1: For \(n\geq 3\) and \(1\leq p\leq 1.0014^{n/4} n^{-3/2}\), \[ R_{n,p}(N_1,N_2)=(p+1)^n D^{-1}_{n,1} D^{-1}_{n,2} \{(2\pi)^{-1} (1-h^2_n)^{-1/2} \exp (-2^{-1} Q(u_1,u_2))+O(n^{-1})\}, \] where \(D^2_{n,1}=np(p+2)/12\), \(D^2_{n,2}=n(n+1)(2n+1)p(p+2)/72\), \(h^2_n=(3n+3)/(4n+2),\) \(u_1=(N_1-2^{-1}np)D^{-1}_{n,1},\) \(u_2=(N_2-4^{-1}n(n+1)p)D^{-1}_{n,2},\) \(Q(u_1,u_2)=(1-h^2_n)^{- 1}(u^2_1-2h_nu_1u_2+u^2_2)\).
By means of a result of Sh. A. Ismatullaev and I. Joó [Acta Math. Hung. 46, 133–149 (1985; Zbl 0617.62012)], Theorem 1 gives information on the distribution of Wilcoxon statistics (Theorem 2).

MSC:

11P81 Elementary theory of partitions
62F05 Asymptotic properties of parametric tests
62G30 Order statistics; empirical distribution functions
11D04 Linear Diophantine equations
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