丢番图方程ax+by=n的一个注记（英文）

令[a,b]为互素的正整数，[n]为非负整数. [D(a,b;n)]表示不定方程[ax+by=n]的非负整数解[(x,y)]的个数. Tripathi证明了[D(a,b;n)=nab+121a+1b+1aj=1a-1ζ-jna1-ζbja+1bk=1b-1ζ-knb1-ζakb]，其中[ζm=e2πi/m]. 在本文中，我们建立了[D(a,b;n)]的递推关系，从而给出了上述结论的新证明.
Let a，b be positive integers such that（a，b）=1 and let n be a non-negative integer. Define [D(a,b;n)] to be the number of non-negative integer solutions（x，y）of the Diophantine equation ax+by=n. Tripathi proved that[D(a,b;n)=nab+121a+1b+1aj=1a-1ζ-jna1-ζbja+1bk=1b-1ζ-knb1-ζakb]，where [ζm=e2πi/m]. In this note，we put forward a recurrence relation of [D(a,b;n)]，thus giving a new proof of above formula