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We update gamma-ray burst (GRB) luminosity relations among certain spectral and light-curve features with 139 GRBs. The distance modulus of 82 GRBs at z>1.4 can be calibrated with the sample at z≤1.4 by using the cubic spline interpolation method from the Union2.1 Type Ia supernovae (SNe Ia) set. We investigate the joint constraints on the Cardassian expansion model and dark energy with 580 Union2.1 SNe Ia sample (z<1.4) and 82 calibrated GRBs’ data (1.4

We present a detailed report of the connection between long-duration gamma-ray bursts (GRBs) and their accompanying supernovae (SNe). The discussion presented here places emphasis on how observations, and the modelling of observations, have constrained what we know about GRB-SNe. We discuss their photometric and spectroscopic properties, their role as cosmological probes, including their measured luminosity–decline relationships, and how they can be used to measure the Hubble constant. We present a statistical summary of their bolometric properties and use this to determine the properties of the “average” GRB-SN. We discuss their geometry and consider the various physical processes that are thought to power the luminosity of GRB-SNe and whether differences exist between GRB-SNe and the SNe associated with ultra-long-duration GRBs. We discuss how observations of their environments further constrain the physical properties of their progenitor stars and give a brief overview of the current theoretical paradigms of their central engines. We then present an overview of the radioactively powered transients that have been photometrically associated with short-duration GRBs, and we conclude by discussing what additional research is needed to further our understanding of GRB-SNe, in particular the role of binary-formation channels and the connection of GRB-SNe with superluminous SNe. [ABSTRACT FROM AUTHOR]