MathDB

For each polynomial

P(x)

with real coefficients, define

P_0=P(0)

and

P_j(x)=x^j\cdot P^{(j)}(x)

where

P^{(j)}

denotes the

j

-th derivative of

P

for

j\geq 1

. Prove that there exists one unique sequence of real numbers

b_0, b_1, b_2, \dots

such that for each polynomial

P(x)

with real coefficients and for each

x

real, we have

P(x)=b_0P_0+\sum_{k\geq 1}b_kP_k(x)=b_0P_0+b_1P_1(x)+b_2P_2(x)+\dots

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