6 When α = 1, this equation reduces to the ordinary Gauss hypergeometric equation. We draw curves of some exam- ple functions in Fig. 7. In this letter, we defined fractional vector calculus in the spherical coordinate framework. We discussed Lapla- cian equation and heat conduct equation in this kind of framework. Some special functions are generalized. The geometry induced by fractional operations is non- addictive and nonlocal. This kind of geometry is not Riemannian; since fractional calculus has found applica- tions in many areas of science and engineering, it is the effective geometry of such physics processes. It will be meaningful to investigate further this kind of geometry. For a complete fractional vector calculus, fractional vector integral is indispensable. But in this letter we re- strained ourselves from this direction. It will be interest- ing to discuss fractional vector integral in our framework and generalize Green’s, Stokes’ and Gauss’s theorem. Special functions show up in different areas of physics and engineering. A fractional generalization of these functions may find applications in similar situations with anomalous tailing behaviors and/or nonlocal properties. the equation (45) can be translated to a fractional differ- ential equation ab · y ( z ) + ( a + b ) z $^{α}$D$^{α}$y ( z ) + z $^{α}$Dα [ z $^{α}$D$^{α}$y ( z ) ] = c · D $^{α}$y ( z ) + z $^{α}$( D $^{α}$) $^{2}$y ( z ) . (48) FIG. 7: Fractional Gauss hypergeometric function. Gauss hypergeometric function is divergent at 1. From the figure we can see this is not the case for fractional equivalences with a bigger α . VI. SUMMARY 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 =1  =4/3  =3/2  =5/3  a=1, b=1, c=1  Fractional Gauss hypergeometric function  [1] I. Podlubny, Fractional Differential Equations , Academic Press, San Diego, 1999. [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Application of Fractional Differential Equations , El- sevier, Amsterdam, 2006. [3] G. M. Viswanathan, T. M. Viswanathan, arXiv:0807.1563v1 [physics.flu-dyn] 10 Jul 2008 [4] V. Gariychuk, B. Datsko, V. Meleshko, Physica A 387 (2008) 418-424 [5] V. Gariychuk, B. Datsko, Physics Letters A 372 (2008) 4902-4904 [6] Lei Song, Shiyun Xu, Jianying Yang, Commun Nonlinear Sci Numer Simulat 15 (2010) 616-628 [7] Nick Laskin, Phys. Rev. E 66, 056108 (2002) [8] Marcelo R. Ubriaco, Physics Letters A 373 (2009) 2516- 2519 [9] Sami I. Muslih, Om P.Agrawal, Int J Theor Phys, DOI: 10.1007/s10773-009-0200-1 [10] Vasily E. Tarasov, Annals of Physics 323 (2008) 2756- 2778 [11] Zhenjun Yan, Mathematical Physics Equations , 2nd Edit., in Chinese, University of Science and Technology of China Press, Hefei, 2001. [12] Zhuxi Wang, Dunren Guo, Introduction to Special Func- tions , in Chinese, Peking University Press, Beijing, 2000. [13] http://en.wikipedia.org/wiki/Hypergeometric function [14] http://en.wikipedia.org/wiki/Confluent hypergeometric function