A note on shifted convolution of cusp-forms with θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-series

In this paper, we study shifted convolution sums for GL(2) and give an upper bound for ∑n≥1λf(n+b)r(n,Q)g(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum \nolimits _{n\ge 1}\lambda _f(n+b) r(n,Q)g(n)$$\end{document}, where g(n) is a smooth weight function. In particular, we get an upper bound for ∑n≤xλf(n+b)r3(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum \nolimits _{n\le x}\lambda _f(n+b)r_3(n)$$\end{document}, which improves the result in Lü et al. (Ramanujan J 40(1):C115–C133, 2016).