Faulhaber's Formula - Umbral Form

Umbral Form

In the classic umbral calculus one formally treats the indices j in a sequence Bj as if they were exponents, so that, in this case we can apply the binomial theorem and say

\sum_{k=1}^n k^p = {1 \over p+1} \sum_{j=0}^p {p+1 \choose j} B_j n^{p+1-j} = {1 \over p+1} \sum_{j=0}^p {p+1 \choose j} B^j n^{p+1-j}


In the modern umbral calculus, one considers the linear functional T on the vector space of polynomials in a variable b given by

Then one can say

\sum_{k=1}^n k^p = {1 \over p+1} \sum_{j=0}^p {p+1 \choose j} B_j n^{p+1-j} = {1 \over p+1} \sum_{j=0}^p {p+1 \choose j} T(b^j) n^{p+1-j}


= {1 \over p+1} T\left(\sum_{j=0}^p {p+1 \choose j} b^j n^{p+1-j} \right) = T\left({(b+n)^{p+1} - b^{p+1} \over p+1}\right).

Read more about this topic:  Faulhaber's Formula

Famous quotes containing the word form:

    “Polarized light showed the secret architecture of bodies; and when the second-sight of the mind is opened, now one color or form or gesture, and now another, has a pungency, as if a more interior ray had been emitted, disclosing its deep holdings in the frame of things.”
    —Ralph Waldo Emerson (1803–1882)