Genus of an entire function

The integer equal to the larger of the two numbers $ p $ and $ q $ in the representation of the entire function $ f ( z) $ in the form

$$ \tag{* } f ( z) = $$

$$ = \ z ^ \lambda e ^ {Q ( z) } \prod _ { k= } 1 ^ \infty \left ( 1 - \frac{z}{a _ {k} } \right ) \mathop{\rm exp} \left ( \frac{z}{a _ {k} } + \frac{z ^ {2} }{2a _ {k} ^ {2} } + {} \dots + \frac{z ^ {p} }{pa _ {k} ^ {p} } \right ) , $$

where $ q $ is the degree of the polynomial $ Q ( z) $ and $ p $ is the least integer satisfying the condition

$$ \sum _ { k= } 1 ^ \infty \frac{1}{| a _ {k} | ^ {p + 1 } } < \infty . $$

The number $ p $ is called the genus of the product appearing in formula (*).

References

[1]	B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1964) (Translated from Russian)

Comments

The genus plays a role in factorization theorems for entire functions, cf. e.g. Hadamard theorem; Weierstrass theorem.

References

[a1]	R.P. Boas, "Entire functions" , Acad. Press (1954)