1 Confluent hypergeometric expansions of the solutions of the double-confluent Heun equation T.A. Ishkhanyan 1,2 and A.M. Ishkhanyan 2 1 Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141700 Russia 2 Institute for Physical Research, NAS of Armenia, 0203 Ashtarak, Armenia Abstract. We present several expansions of the solutions of the double-confluent Heun equation in terms of the Kummer confluent hypergeometric functions. Three different sets of latter functions are examined. It is shown that one type of these functions is applicable only in the case when the double-confluent Heun equation degenerates to the confluent hypergeometric equation, while two other sets of the Kummer functions lead to expansions the convergence radius of which is equal to unity. The coefficients of the expansions in general obey three-term recurrence relations; however a two-term relation is also possible for a particular set of parameters. The conditions for termination of the expansions are discussed. PACS numbers: 02.30.Gp Special functions, 02.30.Hq Ordinary differential equations, 02.30.Mv Approximations and expansions MSC numbers: 33E30 Other functions coming from differential, difference and integral equations, 34B30 Special equations (Mathieu, Hill, Bessel, etc.), 30Bxx Series expansions Keywords: Linear ordinary differential equation, double-confluent Heun equation, special functions, series expansions, three-term recurrence relations The double-confluent Heun equation is one of the four confluent reductions of the general Heun equation [1]. This is a second order linear ordinary differential equation having two irregular singularities of rank 1, each formed by coalescence of a pair of regular singularities of the general Heun equation [2,3]. If the singularities are placed at 0  z and  , the double-confluent Heun equation is written as 0 2 2 2 2             u z q z z d u d z z z d u d     . (1) Note that this form of the equation slightly differs from that adopted in [4], namely, here we have introduced the parameter  instead of the unity used there. This notation is useful in order to distinguish the Whittaker-Ince limit [5] of the double-confluent Heun equation as a particular case achieved within this form by the simple choice 0   . We will see that the notation is also helpful to explicitly reveal that the expansions presented below do not apply to this limiting case. Several expansions in terms of the regular and irregular confluent hypergeometric functions which apply to namely this particular limit are discussed in [6,7].