[Koshkin]

A polynomial always includes a member (monomial) with the power zero. It seems natural, therefore, to index the coefficients correspondingly. In other situations, 1 may be the more natural starting index.

[jhgb]

The useful corner case/"neutral element" for array indexing is the zero-width interval in an arbitrary position.

[xscott]

Fourier coefficients need a zeroth element too.