I am reading a paper here and I've come to a particular passage that is confusing me. It comes on page 2 of the attached paper and it deals with the binomial theorem... The passage lays the groundwork of how umbrals were linear functionals before the spread of linear algebra but the statement that confuses me is at the end of the paragraph and states:

Before knowledge of linear algebra became widespread, the action of a linear functional (written using physicist notation as $\langle L\mid x^n\rangle=a_n$) would be conceived of as raising the index $n$ to a power, and then "treating" the sequence $a_n$ as a sequence of powers $a^n$, while reserving the right to lower the index at the proper time. No precise rules for lowering indices were stated, nor could they be, as long as the underlying conceptual framework was missing. A baffingly difficulty in the calculus of umbrae was the important $rule$ $$(a+a)^n=\sum_{k=0}^{n}\binom{n}{k}a^ka^{n-k}$$ which seemed to imply $a+a\neq 2a$

It is the last statement I am confused at. Why is this implication here? How is it a result?

  • $\begingroup$ That's a really quick proof that the coefficients sum to $2^n$ $\endgroup$
    – David P
    Jan 13, 2015 at 1:05
  • $\begingroup$ I understand the proof. That is easy. But I wasn't understanding the underlying statement answered below. $\endgroup$ Jan 13, 2015 at 3:03

1 Answer 1


I believe the author is writing about confusing the term $a_n$ with the term $a^n$ and vice versa. Interpreted as powers, the binomial theorem is a classical result, but if you interpret the powers as indices, it seems to indicate that the $n$-th component of the sum of a sequence with itself is not the same as twice the $n$-the component of this sequence.

  • $\begingroup$ Now I think I understand where the author is coming from . Thank you. $\endgroup$ Jan 13, 2015 at 3:07

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