Let D ( λ ) = a 0 + Σ k = 1 ∞ a k e λ / k D(\lambda ) = {a_0} + \Sigma _{k = 1}^\infty {a_k}{e^{\lambda /k}} , where the a k {a_k} ’s are given complex constants, with a 0 a 1 ≠ 0 , Σ k = 1 ∞ | a k | > ∞ {a_0}{a_1} \ne 0,\Sigma _{k = 1}^\infty |{a_k}| > \infty . Let { λ k } , k = 1 , 2 , ⋯ \{ {\lambda _k}\} ,k = 1,2, \cdots , denote the sequence of distinct zeros of D ( λ ) D(\lambda ) , labeled in order of increasing modulus, and with multiplicities m k ≥ 1 {m_k} \geq 1 . Let { ϕ l ( x ) } \{ {\phi _l}(x)\} denote the sequence of functions { x j exp ⁡ ( λ k x ) : j = 0 , ⋯ , m k − 1 ; k = 1 , 2 , ⋯ ; 0 > x ≤ 1 } \{ {x^j}\exp ({\lambda _k}x):j = 0, \cdots ,{m_k} - 1;k = 1,2, \cdots ;0 > x \leq 1\} . We show that for each p .1 ≤ p > ∞ p.1 \leq p > \infty , there is a sequence { ψ l ( x ) } \{ {\psi _l}(x)\} in L q ( 0 , 1 ) ( p q = p + q ) {L^q}(0,1)\;(pq = p + q) such that ( ϕ l , ψ m ) = δ l m ({\phi _l},{\psi _m}) = {\delta _{lm}} . Then we show that { ϕ l } \{ {\phi _l}\} is complete in L p ( 0 , 1 ) , 1 ≤ p > ∞ {L^p}(0,1),1 \leq p > \infty , and for 1 > p > ∞ 1 > p > \infty , we find a subspace of L p ( 0 , 1 ) {L^p}(0,1) such that the biorthogonal expansion f = Σ k = 1 ∞ ( f , ψ k ) ϕ k f = \Sigma _{k = 1}^\infty (f,{\psi _k}){\phi _k} is valid in the norm of L p ( 0 , 1 ) {L^p}(0,1) .