Везде далее предполагается $ N=n+m+1 $.

$ n =2, m=1 $:

$$ p(x)=\frac{y_1y_2(x-x_3)(x-x_4)}{(x_1-x_3)(x_1-x_4)(x_2-x_3)(x_2-x_4)}+\frac{y_1y_3(x-x_2)(x-x_4)}{(x_1-x_2)(x_1-x_4)(x_3-x_2)(x_3-x_4)}+ $$ $$ +\frac{y_1y_4(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)(x_4-x_2)(x_4-x_3)}+\frac{y_2y_3(x-x_1)(x-x_4)}{(x_2-x_1)(x_2-x_4)(x_3-x_1)(x_3-x_4)}+ $$ $$ +\frac{y_2y_4(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)(x_4-x_1)(x_4-x_3)}+\frac{y_3y_4(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)(x_4-x_1)(x_4-x_2)} \ , $$ $$ q(x)=\frac{y_1(x_1-x)}{(x_1-x_2)(x_1-x_3)(x_{1}-x_4)}+ \frac{y_2(x_2-x)}{(x_2-x_1)(x_2-x_3)(x_{2}-x_4)}+ $$ $$ +\frac{y_3(x_3-x)}{(x_3-x_1)(x_3-x_2)(x_{3}-x_4)}+ \frac{y_4(x_4-x)}{(x_4-x_1)(x_4-x_2)(x_{4}-x_3)} ; $$

$ n =\forall, m=1 $:

$$ p(x)=\frac{y_1y_2(x-x_3)(x-x_4)\times \dots \times (x-x_N)}{\left\{(x_1-x_3)(x_1-x_4)\times \dots \times (x_1-x_N)\right\} \left\{(x_2-x_3)(x_2-x_4)\times \dots \times (x_2-x_N)\right\} }+ $$ $$ +\frac{y_1y_3(x-x_2)(x-x_4)\times \dots \times (x-x_N)}{\left\{(x_1-x_2)(x_1-x_4)\times \dots \times (x_1-x_N)\right\} \left\{(x_3-x_2)(x_3-x_4)\times \dots \times (x_3-x_N)\right\} }+ \dots ; $$ $$ q(x)=(-1)^n \Bigg[\frac{y_1(x-x_1)}{(x_2-x_1)(x_3-x_1)\times \dots \times (x_{N}-x_1)}+ \frac{y_2(x-x_2)}{(x_1-x_2)(x_3-x_2)\times \dots \times (x_{N}-x_2)}+ \dots \Bigg] $$

$ n =1, m=2 $:

$$ p(x)=\frac{y_1y_2y_3(x-x_4)}{(x_1-x_4)(x_2-x_4)(x_3-x_4)}+\frac{y_1y_2y_4(x-x_3)}{(x_1-x_3)(x_2-x_3)(x_4-x_3)} + $$ $$ +\frac{y_1y_3y_4(x-x_2)}{(x_1-x_2)(x_3-x_2)(x_4-x_2)}+\frac{y_2y_3y_4(x-x_1)}{(x_2-x_1)(x_3-x_1)(x_4-x_1)} \ , $$ $$ q(x)=\frac{y_1y_2(x_1-x)(x_2-x)}{(x_1-x_3)(x_1-x_4)(x_2-x_3)(x_2-x_4)}+\frac{y_1y_3(x_1-x)(x_3-x)}{(x_1-x_2)(x_1-x_4)(x_3-x_2)(x_3-x_4)}+ $$ $$ +\frac{y_1y_4(x_1-x)(x_4-x)}{(x_1-x_3)(x_1-x_2)(x_4-x_3)(x_4-x_2)}+\frac{y_2y_3(x_2-x)(x_3-x)}{(x_2-x_1)(x_2-x_4)(x_3-x_1)(x_3-x_4)}+ $$ $$ +\frac{y_2y_4(x_2-x)(x_4-x)}{(x_2-x_1)(x_2-x_3)(x_4-x_1)(x_4-x_3)}+\frac{y_3y_4(x_3-x)(x_4-x)}{(x_3-x_1)(x_3-x_2)(x_4-x_1)(x_4-x_2)} \ ; $$

$ n =1, m=3 $:

$$ p(x)=\frac{y_1y_2y_3y_4(x-x_5)}{(x_1-x_5)(x_2-x_5)(x_3-x_5)(x_4-x_5)}+\frac{y_1y_2y_3y_5(x-x_4)}{(x_1-x_4)(x_2-x_4)(x_3-x_4)(x_5-x_4)} + $$ $$ +\frac{y_1y_3y_4y_5(x-x_3)}{(x_1-x_3)(x_2-x_3)(x_4-x_3)(x_5-x_3)}+\frac{y_1y_3y_4y_5(x-x_2)}{(x_1-x_2)(x_3-x_2)(x_4-x_2)(x_5-x_2)}+ $$ $$ +\frac{y_2y_3y_4y_5(x-x_1)}{(x_2-x_1)(x_3-x_1)(x_4-x_1)(x_5-x_1)} \ , $$ $$ q(x)=\frac{y_1y_2y_3(x_1-x)(x_2-x)(x_3-x)}{(x_1-x_4)(x_2-x_4)(x_3-x_4)(x_1-x_5)(x_2-x_5)(x_3-x_5)}+\dots $$