函数和它的傅立叶级数之间的关系,常见的有下列四种。命题1 （狄里赫勒定理）若f（x）∈C[-π,π),或在[-π,π]上只有有限个第一类间断点,并且可以把[-π,π]分为f（x）的有限个单调区间,则有f（x）=a0/2+sum from i=1 to ∞（aicosix+bisinix）（1）其中x∈（-π,π）为f（x）的连续点,ai,bi为f（x）的傅立叶系数（以下同）。当x∈（-π,π）为f（x）的间断点时,则（1）式友端改为[f（x—0）+f（x+0）]/2。当x=±π时,则（1）式左端改为[f（-π+0）+f（π-0）]/2。命题2 若f（x）∈L2[-π,π],则对任意确定的n,有||f（x）—a0/2—sum from i=1 to n（a1cosix+bsinix）||2
本文用组合分析的方法及数学归纳法证明了以下一些组合关系式. (1)C(n+k,r)=sum from m=0 to k (k!)/((k-m)!m!)C(n,r-m); (2)sum from m=0 to n K^m C(n,m)=*(1+k)~n; (3)sum from k=0 to n K^m=sum from k=1 to n S(m,k) ((n+1)!)/((k+1)(n-k)!); (4)sum from p=0 to m F(n,p)=((n+m)!)/(n!m!); (5)sum from q=1 to m qF(n,q)=((n+m)!n)/((m-1)!(n+1)!); (6)sum from p=1 to n F(p,m)=((n+m)!)/((m+1)!(n-1)!); (7)sum from r=0 to S (F_(mi2r)F_(n+2r)+F_(m+2r+1)F_(n+2r+1)); =F_(2??+1)(F_(2??+1)F_(m+n+1)+F_(2??)F_(m+n)); (8)sum from k=0 to n C_k=C_(n+5)-2; (9)S_k??5=sum from p=0 to n C_(k+5??)=C_(5n+1+k+γ_(k,5));
