# Problem related to transforming polynomial

Given $$a_1,a_2,…,a_m$$ positive integer. Denominator $$d$$ is smallest positive integer for $$b_l$$ integer coefficient.

$$sum_{k=1}^mbinom{n}k a_k= frac{1}dsum_{l=1}^mb_ln^l$$

Now consider $$n=dt+r$$ where $$d>rge 0$$.

can it be shown that, above equation transform as

$$frac{1}dsum_{l=1}^mb_l(dt+r)^l=sum_{u=0}^{m-1}left(frac{x_ut+y_u}{z_u}right)(dt+r)^u$$

With $$x_u,y_u$$ and $$z_u$$ integers.

Example

Let $$a_1=1,a_2=1,a_3=3$$

then $$sum_{k=1}^3binom{n}k a_k=(n^3-2n^2+3n)/2$$ here $$d=2$$

Case$$(1)$$ $$n=2t$$,

$$frac{n^3-2n^2+3n}2=(t-1)n^2+n+t$$

Case$$(2)$$ $$n=2t+1$$,

$$frac{n^3-2n^2+3n}2=(t-1)n^2+(t+2)n$$

Thank you.