Parity Patterns Under Power-Series Operations

Why Parity Patterns Matter

A power series is not only a way to represent a function. Its coefficient pattern can also reveal structural features of that function. In particular, a series containing only even powers is symmetric about its center, while one containing only odd powers changes sign across its center.

Knowing how these patterns behave under algebraic and calculus operations makes it possible to predict the form of a result before calculating every coefficient. The center matters: parity here is measured relative to the center of the series, not necessarily relative to \(x=0\).

Parity About the Center

Write a power series centered at \(c\) as \(\sum_{n=0}^{\infty}a_n(x-c)^n\), with radius of convergence \(R>0\). For \(|x-c|<R\), it represents a function \(f\). It is useful to set \(t=x-c\), so the powers being considered are powers of \(t\).

The constant term is allowed in an even series but not in an odd series. The zero series satisfies both coefficient conditions. For a power series, the symmetry descriptions agree with the coefficient descriptions: substituting \(-t\) multiplies the coefficient of \(t^n\) by \((-1)^n\), and equality of power-series coefficients determines which terms must vanish.

Addition: Matching Patterns Stay Matching

Two power series with the same center can be added coefficient by coefficient wherever both converge. If both are even, their odd-indexed coefficients are zero, so the sum is even. If both are odd, their even-indexed coefficients are zero, so the sum is odd.

Adding series of different parity does not generally produce a series of either parity. The even terms come from the even series and the odd terms from the odd series. A special cancellation or a zero summand can change that conclusion, so the coefficients should be checked rather than assuming a general rule.

Multiplication: Parity Combines Like Signs

For power series \(A(t)=\sum_{n=0}^{\infty}a_nt^n\) and \(B(t)=\sum_{n=0}^{\infty}b_nt^n\), their product has Cauchy product coefficients \(d_n=\sum_{k=0}^{n}a_kb_{n-k}\), wherever the product-series operation is valid. In this finite sum, a nonzero contribution to degree \(n\) can occur only by multiplying a term of degree \(k\) by one of degree \(n-k\).

If both factors are even, only even \(k\) and even \(n-k\) can contribute, so \(n\) must be even. If both are odd, both degrees are odd, and their sum is even. If one factor is even and the other odd, each contributing pair has one even degree and one odd degree, so the total degree is odd.

Differentiation Reverses Parity

Termwise differentiation is valid inside the radius of convergence of a power series. If \(f(c+t)=\sum_{n=0}^{\infty}a_nt^n\), then its derivative with respect to \(x\), noting that \(t=x-c\), is represented there by \(\sum_{n=1}^{\infty}n a_n t^{n-1}\).

For an even series, every nonconstant term has even degree \(n\), so differentiation produces degree \(n-1\), which is odd. The constant term differentiates to zero. For an odd series, every term has odd degree \(n\), and \(n-1\) is even, so its derivative is even. Thus differentiation reverses parity, with a constant even series as the case whose derivative is zero.

Integration Reverses Parity, With a Constant to Track

Termwise integration is valid inside the radius of convergence. Integrating a term \(a_nt^n\) gives \(a_nt^{n+1}/(n+1)\), and an indefinite integral also has an arbitrary constant. Since the exponent increases by one, an even integrand produces an odd power-series part, while an odd integrand produces an even power-series part.

The constant of integration is important. An odd series integrated term by term gives an even series of nonconstant powers, but adding a constant keeps the result even. An even series integrated term by term gives an odd series; adding a nonzero constant generally makes the antiderivative neither even nor odd. The parity claim for an antiderivative therefore depends on the chosen constant.

Why the Rules Work

The coefficient rules can be justified directly, without relying only on examples. Let \(t=x-c\). For addition, the coefficient of \(t^n\) in a sum is the sum of the two coefficients at degree \(n\). If both series are even, those coefficients vanish for every odd \(n\); if both are odd, they vanish for every even \(n\). This proves the addition rules.

For products, the coefficient of degree \(n\) is the finite convolution \(\sum_{k=0}^{n}a_kb_{n-k}\). In an even-even product, a nonzero summand requires \(k\) and \(n-k\) both even, making \(n\) even. In an odd-odd product, both indices must be odd, again making \(n\) even. In an even-odd product, one index must be even and the other odd, making \(n\) odd. Every coefficient of the other parity is therefore zero.

These calculations prove the coefficient-pattern claims. They apply to power series on the interiors where the relevant operations are valid: addition and Cauchy products where the series represent their sum and product, and termwise differentiation or integration within the series' radius of convergence.

A Common Pitfall: Forgetting the Center or the Constant

Parity is about powers of \(x-c\), not automatically powers of \(x\). For example, \(x-1\) is odd about \(c=1\), because it equals \(t\), but it is not an odd function about zero: replacing \(x\) by \(-x\) gives \(-x-1\), which is not \(-(x-1)\).

A second error is to claim that every antiderivative of an even function is odd. For instance, \(2+3(x-1)^2\) is even about \(1\), but one antiderivative is \(2(x-1)+(x-1)^3+4\). The added constant means this expression is generally neither even nor odd about \(1\). Differentiating it confirms the original integrand, but does not remove the constant from the antiderivative. To obtain an odd antiderivative in this example, choose \(C=0\).