<p>Recently, Huang and Shen (SIAM J NUMER 62.4, 1609–1637) construct a new class of BDF schemes for the parabolic type equations based on the Taylor expansions at time <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(t^{n+\beta }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <mi>β</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\beta &gt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>&gt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> being a tunable parameter. These new BDF schemes allow larger time steps at higher order than that which allowed with a usual higher-order scheme. Inspired by their excellent work, we construct in this paper a series of new generalized BDF (nGBDF) schemes and its accelerated version for the parabolic type equations. The newly considered schemes not only maintain all the advantages of the original schemes, but more importantly further expand the stability regions. Furthermore, an accelerated technique is proposed to greatly improve the computational speed of the schemes. Besides, we also conduct rigorously error analysis for the second-order and the third-order nGBDF schemes. Finally, various numerical examples are considered to support our findings.</p>