Construction of cubic spline hidden variable recurrent fractal interpolation function and its fractional calculus.

• We calculate calculus of hidden variable recurrent fractal interpolation function and give a theorem on the existence of Cr -HVRFIF. • We construct CSHVRFIFs of Type-I, Type-II and cardinal CSHVRFIFs and estimate error bound between the constructed CSHVRFIF and an original function. • We show that fractional calculus of cardinal CSHVRFIFs are also HVRFIFs. In this paper, we construct a cubic spline hidden variable recurrent fractal interpolation function(CSHVRFIF) and prove that its Riemann-Liouville fractional integral and derivative are also HVRFIFs. To do it, firstly, we calculate calculus of hidden variable recurrent fractal interpolation function(HVRFIF) and give a theorem on the existence of Cr -HVRFIF. Secondly, we construct CSHVRFIFs of Type-Ⅰ and Type-Ⅱ, cardinal CSHVRFIFs and estimate error bound between the constructed CSHVRFIF and an original function. Finally, we insist that fractional calculus of cardinal CSHVRFIFs are also HVRFIFs. [ABSTRACT FROM AUTHOR]