The Asymptotic Expansion of a Function Introduced by L.L. Karasheva

The asymptotic expansion for x→±∞ of the entire function Fn,σ(x;μ)=∑k=0∞sin(nγk)sinγkxkk!Γ(μ−σk),γk=(k+1)π2n for μ>0, 0<σ<1 and n=1,2,… is considered. In the special case σ=α/(2n), with 0<α<1, this function was recently introduced by L.L. Karasheva (J. Math. Sciences, 250 (2020) 753–759) as a solution of a fractional-order partial differential equation. By expressing Fn,σ(x;μ) as a finite sum of Wright functions, we employ the standard asymptotics of integral functions of hypergeometric type to determine its asymptotic expansion. This was found to depend critically on the parameter σ (and to a lesser extent on the integer n). Numerical results are presented to illustrate the accuracy of the different expansions obtained.


Introduction
In a recent paper, L.L. Karasheva [1] introduced the entire function , where µ > 0, 0 < α < 1 and n = 1, 2, . . . and, throughout, x is a real variable. This function is of interest as it is involved in the fundamental solution of the differential equation for positive integer n, where the derivative with respect to t is the fractional derivative of the order α. In the simplest case n = 1, we have Θ 1,α (x; µ) = φ(−σ, µ; x), σ := α/(2n), where φ(−σ, µ; x) is the Wright function which finds application as a fundamental solution of the diffusion-wave equation [2]. Under the above assumptions on n and α it follows that the parameter σ associated with (1) satisfies 0 < σ < 1 2 . In this study, however, we shall allow the parameter σ to satisfy 0 < σ < 1 and consider the function which coincides with Θ n,α (x; µ) when σ = α/(2n). From the well-known expansion where ω r := (n − 2r − 1)π 2n (0 ≤ r ≤ n − 1), (4) it follows that (3) can be expressed as a finite sum of Wright functions defined in (2) with rotated arguments (compare [1], Equation (4)) We note that the extreme values of ω r satisfy ω 0 = −ω n−1 = (n − 1)π/(2n), whence |ω r | < 1 2 π for 0 ≤ r ≤ n − 1. We use the representation in (5), with the values of ω r in (4), to determine the asymptotic expansion of F n,σ (x; µ) for x → ±∞ by application of the asymptotic theory of the Wright function. A summary of the expansion of φ(−σ, µ; z) for large |z| is given in Section 3. The expansions of F n,σ (x; µ) for x → ±∞ are given in Sections 4 and 5, where they are shown to depend critically on the parameter σ (and to a lesser extent on the integer n). A concluding section presents our numerical results confirming the accuracy of the different expansions obtained.

An Alternative Representation of F n,σ (x; µ)
The Wright function appearing in (2) can be written alternatively as upon use of the reflection formula for the gamma function, where ϑ := 1 2 − µ. The associated Wright function Ψ(z) is defined by which is valid for |z| < ∞. Hence, we obtain the representation If we now exploit the symmetry of the ω r in (4) (and the fact that x is a real variable), we observe that the values of ω r for 0 ≤ r ≤ N − 1, where N = n/2 , satisfy , . . . , π 2n n , n = 1 (n even) 2 (n odd).
The form (8) involves half the number of Wright functions Ψ(z) and will be used to determine the asymptotic expansion of F n,σ (x; µ) as x → ±∞ in Sections 4 and 5.

The Asymptotic Expansion of Ψ(z) for |z| → ∞
We first present the large-|z| asymptotics of the function Ψ(z) in (6) based on the presentation described in ( [3], Section 4); see also ([4], Section 4.2), ( [5], §2.3). We introduce the following parameters: together with the associated (formal) exponential and algebraic expansions where (The dependence of the coefficients A j (σ) on the parameter δ is not indicated.) Then, since 0 < κ < 1, we obtain from ( [5], p. 57) the large-z expansion where the upper or lower signs are chosen according as arg z > 0 or arg z < 0, respectively. The expansion E(z) is exponentially large as |z| → ∞ in the sector | arg z| < 1 2 πκ, and oscillatory (multiplied by the algebraic factor z ϑ/κ ) on the anti-Stokes lines arg z = ± 1 2 πκ. In the adjacent sectors 1 2 πκ < | arg z| < πκ, the expansion E(z) continues to be present, but is exponentially small reaching maximal subdominance relative to the algebraic expansion on the Stokes lines (On these rays, E(z) undergoes a Stokes phenomenon where it switches off in a smooth manner (see [6], p. 67).) arg z = ±πκ. In our treatment of F n,σ (x; µ), we will not be concerned with exponentially small contributions, except in one special case when x → −∞ where the expansion of F n,σ (x; µ) is exponentially small.
The first few normalised coefficients c j = A j (σ)/A 0 (σ) are [3,4]: In addition to the Stokes lines arg z = ±πκ, where E(z) is maximally subdominant relative to the algebraic expansion, the positive real axis is also a Stokes line. Here, the algebraic expansion is maximally subdominant relative to E(z). As the positive real axis is crossed from the upper to the lower half plane the factor e −πi appearing in H(ze −πi ) changes to e πi , and vice versa. The details of this transition will not be considered here; see ( [5], p. 248) for the case of the confluent hypergeometric function 1 F 1 (a; b; z).

. Asymptotic Character as a Function of σ
Let us denote the arguments of the Ψ functions appearing in (8) by The representation of the asymptotic structure of the functions Ψ(z ± r ) is illustrated in Figure 1 for different values of σ. The figures show the rays arg z = ±πσ and the anti-Stokes lines (dashed lines) arg z = ± 1 2 πκ. In the case σ = 2 3 , the exponentially large sector is | arg z| < 1 6 π, and it is seen from Figure 1a that the arguments z ± r for 0 ≤ r ≤ N − 1 and xe ±πiσ all lie in the domain where Ψ(z) has an algebraic expansion; this conclusion applies a fortiori when 2 3 < σ < 1. When σ = 1 2 , the exponentially large sector is | arg z| < 1 4 π; when n = 2, we have ω 0 = 1 4 π so that z + 0 is situated on the boundary of the exponentially large sector.
Other values of n ≥ 3 will have some z + r inside this sector, whereas the z − r are in the algebraic sector for n ≥ 2. Similarly, the case σ = 1 3 , where the rays arg z = ±πσ and arg z = ± 1 2 πκ coincide, has all the z + r situated in the exponentially large sector, with the z − r situated in the algebraic domain. Finally, when σ = 1 6 , the exponentially large sector | arg z| < 5 12 π encloses the rays arg z = ±πσ with the result that all the z + r lie in the exponentially large sector, whereas the z − r lie in the algebraic domain (except when n = 2 when z − 0 lies on the lower boundary of the exponentially large sector).
To summarise, we have the following asymptotic character of F n,σ (x; µ) when x → +∞ as a function of the parameter σ: Outside the exponentially large sector, the expansion of Ψ(z) is algebraic in character. The circular quadrants represent the range of the arguments arg z = ±πσ − ω r for 0 ≤ r ≤ n/2 − 1, with n ≥ 2 and the arrow-head corresponds to n = ∞. When σ = 1/3, the rays arg z = ±πσ and arg z = ± 1 2 πκ coincide.
To summarise, we have the following asymptotic character of F n,σ (x; µ) when x → +∞ as a function of the parameter σ:

Karasheva's Estimate for |Θ n,α (x; µ)|
When σ = α/(2n) < 1 2 , we see from Theorem 1 that the dominant exponential expansion as x → +∞ corresponds to r = 0, yielding Thus, we have the leading order estimate as x → +∞. When expressed in our notation, Karasheva's estimate for |Θ n,α (x; µ)| in ( [1], §8) agrees with (20) (when the second cosine term is replaced by 1), except that she did not give the value of the multiplicative constant A 0 (σ)/π given in (11). However, the presentation of her result as an upper bound is not evident due to the presence of possibly less dominant exponential expansions and also the subdominant algebraic expansion.

The Expansion of F n,σ (x; µ) for x → −∞
To examine the case of negative x, we replace x by e ∓πi x, with x > 0, and use the fact that Ψ(ze 2πi ) = Ψ(z) to find, from (8), that The rays arg z = ±πσ in Figure 1 are now replaced by the Stokes lines arg z = ±πκ. The Stokes and anti-Stokes lines arg z = ± 1 2 πκ are illustrated in Figure 2 when 0 < σ < 1 2 and 1 2 < σ < 1. In the sectors 1 2 πκ < | arg z| < πκ, we recall that the exponential expansion E(z) is still present but is exponentially small as |z| → ∞.  Figure 2. Diagrams representing the rays arg z = ±πκ and the boundaries of the exponentially large sector (shown by dashed rays) | arg z| < 1 2 πκ, κ = 1 − σ for (a) 0 < σ < 1 2 and (b) 1 2 < σ < 1. The circular quadrants represent the range of the arguments arg z = ±πκ − ω r for 0 ≤ r ≤ N − 1 with the arrow-head corresponding to n = ∞. The ± signs in (b) denote the signs to be chosen in H(z) on either side of the Stokes line arg z = 0.
For the algebraic component of the expansion two cases arise when the argument πκ − ω r of the second Ψ function in Υ r (−κ; x) is either (i) positive or (ii) negative. In case (i) the algebraic expansion H(z) does not encounter a Stokes phenomenon as its argument does not cross arg z = 0, whereas in case (ii) a Stokes phenomenon arises for those values of r that make πκ − ω r < 0. In case (i), the algebraic component contains the factor inside the sum over r in (21) e πiϑ (e −πiκ−iω r · e πi ) −K + e −πiϑ (e πiκ−iω r · e −πi ) −K Figure 2. Diagrams representing the rays arg z = ±πκ and the boundaries of the exponentially large sector (shown by dashed rays) | arg z| < 1 2 πκ, κ = 1 − σ for (a) 0 < σ < 1 2 and (b) 1 2 < σ < 1. The circular quadrants represent the range of the arguments arg z = ±πκ − ω r for 0 ≤ r ≤ N − 1 with the arrow-head corresponding to n = ∞. The ± signs in (b) denote the signs to be chosen in H(z) on either side of the Stokes line arg z = 0.
For case (ii) to apply, we require that πκ − ω 0 < 0; that is, n > n * = 1/(2σ − 1). Suppose that πκ − ω r < 0 for 0 ≤ r ≤ r 0 . Then, the algebraic component resulting from the terms with r ≤ r 0 becomes where, in the second term in round braces, we have taken account of the Stokes phenomenon (the first term and that multiplied by ∆ n are unaffected). Some routine algebra then produces the algebraic contribution when n > n * andĤ(x) ≡ 0 when n < n * . (We avoid here consideration of the algebraic contribution when πκ − ω r = 0, that is, on the Stokes line arg z = 0.) Reference to Figure 2 shows that there is no exponential contribution to F n,σ (−x; µ) from the terms Ψ(xe −πiκ ) and Ψ(xe −πiκ−iω r ). From (10) and (21), we find the exponential expansion results from the terms Ψ(xe πiκ−iω r ), which is given bŷ where X and the asymptotic sum S are defined in (17) and (18) with Φ := ω r (1 + ϑ/κ). For σ < 1 2 (when the algebraic expansion vanishes), the expansion of F n,σ (−x; µ) will be exponentially small provided πκ − ω 0 > 1 2 πκ; that is, when n < 1/σ. If n = 1/σ, there is an exponentially oscillatory contribution, and when n > 1/σ, the expansion is exponentially large.

Numerical Results
In this section, we describe numerical calculations that support the expansions given in Theorems 1 and 2. The function F n,σ (x; µ) was evaluated using the expression in terms of Wright functions (valid for real x) which follows from (5) and the symmetry of ω r .
In Table 1, we present the results of numerical calculations for x → +∞ compared with the expansions given in Theorem 1. We choose four representative values of σ that focus on the different cases of Theorem 1 and n = 2, 3 and 4. The numerical value of F n,σ (x; µ) was obtained by high-precision evaluation of (25). The exponential expansion E(x) was computed with the truncation index j = 3 and the algebraic expansion H(x) was optimally truncated (that is, at or near its smallest term).
The first case σ = 1 3 has an exponentially large expansion with a subdominant algebraic contribution for all three values of n. The second case σ = 1 2 corresponds to n 0 = 2; when n = 2, E(x) is oscillatory and makes a similar contribution as H(x), whereas when n = 3 and 4, E(x) is exponentially large. The third case σ = 5 9 corresponds to n 0 = 3; when n = 2, there is no exponential contribution, whereas when n = 3, E(x) is oscillatory and thus makes a similar contribution as H(x); when n = 4, E(x) is exponentially large. Finally, when σ = 2 3 , the expansion of F n,σ (x; µ) is purely algebraic in character. In Table 2, we present illustrative examples of Theorem 2 when x → −∞. The first case, σ = 1 4 (κ = 3 4 ), has an expansion that is exponential in character; for n < 1/σ = 4, E(x) is exponentially small, whereas for n = 4, the argument πκ − ω 0 = 3 8 π lies on the upper boundary of the exponentially large sector | arg z| < 3 8 π, and thusÊ(x) is oscillatory. For n ≥ 5,Ê(x) becomes exponentially large as x → −∞. In the second case, σ = 2 5 (κ = 3 5 ), E(x) is exponentially small for n = 2 and exponentially large for n ≥ 3.

Concluding Remarks
We employed the standard asymptotics of the Wright function Ψ(z) defined in (6) to determine the asymptotic expansion of F n,σ (x; µ) for x → ±∞. We found that this behaviour depended critically on the parameter σ. The numerical results presented in Tables 1 and 2 demonstrate that the asymptotic forms of F n,σ (x; µ) stated in Theorems 1 and 2 agreed well with the numerically computed values of F n,σ (±x; µ). In particular, we showed that, when σ < 1 2 , the expansion of F n,σ (x; µ) exponentially decays as x → −∞.