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Let $X^{(n,m)}_{(1,d)}$ denote the Segre-Veronese embedding of $\mathbb{P}^n\times \mathbb{P}^m$ via the sections of the sheaf $\mathcal{O}(1,d)$. We studythe dimensions of higher secant varieties of $X^{(n,m)}_{(1,d)}$ and we provethat there is no defective $s^{th}$ secant variety, except possibly for $n$values of $s$. Moreover when ${m+d \choose d}$ is multiple of $(m+n+1)$, the$s^{th}$ secant variety of $X^{(n,m)}_{(1,d)}$ has the expected dimension forevery $s$.

Higher secant varieties of $\mathbb{P}^n \times \mathbb{P}^m$ embedded in bi-degree $(1,d)$