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SOLUTION OF DIFFERENTIAL EQUATIONS OF HYPERGEOMETRIC TYPE J. A. PALMER

Abstract. We present a method for solving the classical linear ordinary differential equations of hypergeometric type [8], including Bessel’s equation, Legendre’s equation, and others with polynomial coefficients of a certain type. The method is characterized by using the Mellin transform to convert the original differential equation into a complex difference equation solving the differential equations in the Mellin transform, or log-spectral, domain.

1. Introduction The standard method of solving differential equations with variable coefficients is the series method of Frobenius. If certain order criteria apply to the singularities, then a series solution can be generated. Two generate an independent This paper considers the solution of differential equations of hypergeometric type. The proposed method is based on using the Mellin transform to convert the differential equation into a first order difference equation in the complex domain, for which solutions can be constructed using Γ-functions. We propose that the common feature of functions of hypergeometric type1 is this property of yielding a first order complex difference equation. The Mathieu equation, for example, yields a second order difference equation, which is not solvable by the proposed method. This corresponds to its being of “higher type” than the functions of hypergeometric type [8, §19.1]. An advantage of the proposed method over series methods like that of Frobenius, is that multiple linearly independent solutions are found simultaneously, and may be represented in the original domain either in series form, or as the scale convolution of two or more constituent functions. A notable case in which a second solution cannot be found using the Frobenius method is Bessel’s equation of integer order. Bessel’s function of the second kind is commonly defined and then verified to be an independent solution and satisfy Bessel’s equation. Similarly for Legendre’s function of the second kind. Our method yields the second solutions, plus others, simultaneously with the first. Furthermore, each solution is represented in a schematic form that can be seen as being central in the sense that and Our method differs from that used by Mellin [?, 3, §7.6] in our emphasis on the difference equation and the simultaneous generation of multiple independent solutions for various boundary conditions. The differen [2, §7.6] by Hille . Hille, however, seems to treat the method more as a peculiarity [2, p. 279] than a general method, using the Frobenius method in the main treatment of differential equations. 1There is evidently no standard definition [4, p. 56] of hypergeometric type. c

1997 American Mathematical Society

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J. A. PALMER

Again, Bessel’s equation of the second kind is defined and verified to be a solution of Bessel’s equation, rather than derived on an equal footing with other solutions. In § 2, we review the basic properties and inverse theorems for the Mellin transform. In § 3 we demonstrate the method for solving some classical equations. 2. The Mellin transform The Mellin transform [7, 5, 1], is defined by, Z ∞ (1) M [f (x) ; s] ≡ f˜(s) ≡ xs−1f (x) dx 0

for s ∈ C such that the integral is convergent. 2.1. Basic properties. If xk g(x) and xk h(x) ∈ L(0, ∞) for some k ∈ R, then [7, Thm. 44], Z ∞ 1 x ˜ (2) f (x) = g h(ξ) dξ ⇒ f˜(s) = g˜(s)h(s) ξ 0 ξ and xkf (x) ∈ L(0, ∞). Thus under appropriate conditions, we can solve for h(x) by inverting the transform. If xkf (x) ∈ L(0, ∞), then (1) can be inverted almost everywhere using the formula [7, Thm. 28], (3)

M

−1

[f˜(s) ; x] ≡

1 2πi

k+i∞ Z

x−s f˜(s) ds =

f (x+) + f (x−) 2

k−i∞

where f (x+) and f (x−) denote the right and left hand limits of f at x. Thus, for example, when h in (2) is continuous on (0, ∞), we have, f˜(s) ; x], x ∈ (0, ∞) h(x) = M−1 [ g˜(s) R∞ We can similarly solve integral equations of the form f (x) = 0 g(ξx)h(ξ) dξ. Using the following two properties, which follow readily from the definition (1), 1˜ s a (4) M [x f (x) ; s] = f , M [f (xa ) ; s] = f˜(s + a) a a ˜ − s), and it follows that, we have M t−1 h(t−1 ) ; s = h(1 Z ∞ ˜ − s) (5) f (x) = g(ξx)h(ξ) dξ ⇒ f˜(s) = g˜(s)h(1 0

Like the Laplace transform, the Mellin transform can be used to convert integrodifferential equations into algebraic equations. We use in particular the following relation. Let D denote the differential operator. The transform of the operator (−x)n Dn is given by, (6)

M [(−x)nDnf (x) ; s] =

Γ(s + n) ˜ f (s) Γ(s)

Rx R∞ There are two basic integrals that are used, 0 f (x)dx and x , and there are two corresponding definitions of fractional integrals. The Riemann-Liouville fractional integral [1, p. 113], for α > 0 and non-integral, is defined by, Z x 1 −α D f (x) = (t − x)α−1 f (t) dt Γ(α) 0

SOLUTION OF DIFFERENTIAL EQUATIONS OF HYPERGEOMETRIC TYPE

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The Riemann-Liouville fractional derivative, Dβ , β > 0, is defined by the same formula, with α replaced by −β. The Weyl fractional integral, for α > 0 and non-integral, is defined by, Z ∞ 1 W −α f (x) = (t − x)α−1 f (t) dt Γ(α) x For the Weyl fractional derivative, W β , β > 0, let n be the smallest integer greater than β. Then the Weyl fractional derivative is defined by, W β f (x) = (−D)n W n−β f (x) The Mellin transform of the Weyl fractional integral or derivative (α positive or negative), is given by, Γ(s) ˜ (7) M W −α f (x) ; s = f (s + α) Γ(s + α) We also use the following result on differentiation in the transform domain. dn ˜ (8) M [(log x)n f (x) ; s] = f (s) dsn 2.2. Inversion theorems. When f˜(σ + it) decays exponentially as |t| → ∞, the Mellin transform can be inverted using the following theorem [7, Thm. 31], Theorem 1. Let s = σ + it and z = reiθ . The function f˜(s) is given by, Z ∞ f˜(s) = z s−1 f (z) dz 0

and has the following two properties, (S1) f˜(s) = f˜(σ + it) is analytic in the strip, a < σ < b , (S2) |f˜(σ + it)| = O(e(α+)t ) as t → −∞, and O(e−(β−)t ) as t → ∞, (with 0 < α, β ≤ π) uniformly in any closed strip interior to a < σ < b , if and only if, the function f (z) is given by, k+i∞ Z

z −s f˜(s) ds

f (z) = k−i∞

for any a < k < b, and has the two properties, (Z1) f (z) = f (reiθ ) is analytic in the wedge −α < θ < β, (Z2) |f (reiθ )| = O(r−a− ) as r → 0, and O(r−b+ ) as r → ∞, uniformly in any closed wedge interior to −α < θ < β . If f˜(s) does not decay exponentially as |t| → ∞, then we require another theorem for the inverse. R∞ Theorem 2. Let the integral 0 xs−1f (x) dx ≡ f˜(s) be uniformly convergent (as the limits are approached independently) for s = k + it, t in any finite interval. Then, k+iλ Z 1 |t| −s ˜ f (x+) + f (x−) 1− (9) lim x f (s) ds = 2πi λ→∞ λ 2 k−iλ

for all x > 0 such that f (x+) and f (x−) exist.

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J. A. PALMER

The asymptotic order of the Gamma function can be determined from Stirling’s formula. Specifically [6, §4.42], for fixed θ, we have, Γ(reiθ )

1 = O z z− 2 e−z 1

= O rr cos θ− 2 e−r cos θ−r θ sin θ

as r → ∞. And for fixed σ, Γ(σ + it)

1 = O ss− 2 e−s 1

π

= O |t|σ− 2 e− 2 |t|

as |t| → ∞, and,

3. Differential equations of hypergeometric type 3.1. Classical equations. Bessel’s differential equation is defined by, (10)

x2

d 2f df +x + (x2 − ν 2 )f = 0 dx2 dx

Taking Mellin transforms, we have, s(s + 1)f˜(s) − sf˜(s) + f˜(s + 2) − ν 2 f˜(s) = 0 or, rearranging, f˜(s + 2) = (ν 2 − s2 )f˜(s)

(11)

That is, when f˜(s) is shifted to the left by 2, two simple zeros are introduced, one at ν and one at −ν. Now consider the hypergeometric differential equation, which is defined by, x(1 − x)

df d 2f + c − (a + b + 1) x + abf = 0 dx2 dx

Taking Mellin transforms and rearranging, we get, (12)

(s − a)(s − b) ˜ f˜(s − 1) = f (s) (s − c)(s − 1)

Here, shifting f˜(s) to the right by 1 introduces zeros at a and b, and poles at c and 1. The following table lists the classical second order differential equations that give rise to first order complex difference equations,

SOLUTION OF DIFFERENTIAL EQUATIONS OF HYPERGEOMETRIC TYPE

Name

Differential Equation

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Interval

Bessel

x2 y 00 + xy 0 + (x2 − ν 2 )y = 0

(0, ∞)

Modified Bessel

x2 y 00 + xy 0 − (x2 + ν 2 )y = 0

(0, ∞)

Spherical Bessel

x2 y 00 + 2xy 0 + (x2 − ν(ν + 1))y = 0

(0, ∞)

Struve

x2 y 00 + xy 0 + (x2 − ν 2 )y =

Legendre

(1 − x2 )y 00 − 2xy 0 + n(n + 1)y = 0

(−1, 1)

Chebyshev I

(1 − x2 )y 00 − xy 0 + n2 y = 0

(−1, 1)

Chebyshev II

(1 − x2 )y 00 − 3xy 0 + n(n + 2)y = 0

(−1, 1)

Laguerre

xy 00 − (1 − x)y 0 + ny = 0

(0, ∞)

Confluent hypergeometric

xy 00 + (c − x)y 0 − ay = 0

−

Hermite

y 00 − 2xy 0 + 2ny = 0

Hypergeometric

x(1 − x)y 00 + (c − (a + b + 1)x)y 0 + ab y = 0

21−ν √ Γ(ν+ 1 ) π 2

xν+1

(0, ∞)

(−∞, ∞) −

The corresponding Mellin difference equations are given in the following table. Type

Complex difference equation

Bessel

y˜(s + 2) = ±(s − ν)(s + ν)˜ y (s)

Legendre

y˜(s − 2) =

Confluent hypergeometric y˜(s − 1) =

(s − n)(s − (n + 1)) y˜(s) (s − 1)(s − 2) (s − a) y˜(s) (s − c)(s − 1)

Hermite

y˜(s − 2) = −2

Hypergeometric

y˜(s − 1) =

(s + n) y˜(s) (s − 1)(s − 2)

(s − a)(s − b) y˜(s) (s − c)(s − 1)

In each case, we have reduced the problem to the determination of a complex function, integrable in some strip, satisfying, (13)

F (s + δ) =

p (s) F (s) q(s)

where p(s) and q(s) are polynomials with real zeros. The basic idea of the proposed method is to construct meromorphic solutions to (13) by appropriately combining Γ-functions.

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J. A. PALMER

For our purposes, a Γ-function, say Γ((s − a)/b), a, b ∈ R, is regarded as an infinite sequence of poles beginning at a, extending in the direction −sgn(b) and evenly spaced at a distance of |b|. Similarly, the function cos(π(s − a)/b) is viewed as an infinite sequence of zeros extending in the positive and negative directions. The classical formulae relating the trigonometric and Γ functions are easily deduced by viewing the functions this way and matching poles and zeros. π π = Γ(s)Γ(1 − s) = Γ( 12 + s)Γ( 12 − s) sin(πs) cos(πs) 3.2. Difference equations. Suppose we wish to construct a function that satisfies, F (s + δ) = (s − a)F (s)

(14)

where δ ∈ R. That is, shifting F (s) to the left by δ introduces a simple zero at a. The Γ-function is the prototype of functions of this kind, satisfying, Γ(s+1) = sΓ(s). Let us first note that if F (s) satisfies (14), then F (s) cos(2π(s − r)/δ) and

F (s) cos(2π(s − r)/δ)

also satisfy (14) for any r ∈ R. We can use this type of cosine term to change the asymptotic order of F (s) to make it integrable. Also note that if F satisfies, (15)

F (s + δ) = G(s)F (s)

and we define F¯ by, F¯ (s) ≡ F (s) cos(π(s − r)/δ) then F¯ satisfies, F¯ (s + δ) = −G(s)F¯ (s)

(16)

for any r ∈ R. This type of cosine term can be used when it is desired to change the sign in the functional equation satisfied by F (s). We shall denote these cosine terms by, (17) C1 (r) ≡ cos π(s − r)/δ C2 (r) ≡ cos 2π(s − r)/δ The δ will generally be taken to be understood from the equation being considered, and we assume that the functional equation is shifted to make δ positive. As a final preliminary observation, note that if F (s) satisfies (15), and we define, F¯ (s) ≡ αs/δ F (s) with α > 0, then F¯ satisfies, F¯ (s + δ) = α G(s)F¯ (s) Now, in order to construct a function such that a shift introduces a zero at a, we have two basic options: either shift the end of a one-sided sequence of zeros onto a, or shift the end of a one-sided sequence of poles off of a. Likewise, to introduce a pole at a, we can either shift a sequence of poles onto a, or shift a sequence of zeros off of a. Specifically, to construct a function satisfying (14), we can use either of, (18)

A1 (a) ≡ δ s/δ Γ (s − a)/δ

A2 (a) ≡

δ s/δ Γ (a + δ − s)/δ cos(π(s − r)/δ)

where the cosine in A2 is used to turn a − s into s − a, as in (16). The following figure illustrates the idea. In the cosine term in A2 , r is taken to be a.

SOLUTION OF DIFFERENTIAL EQUATIONS OF HYPERGEOMETRIC TYPE

··· ···

x

x

A1

x

x

x

b

x

x

b A2 b

b

7

x ··· ···

x a

a+δ

When A1 or A2 is shifted to the left by δ, the effect is to introduce a simple zero at a. Similarly, to introduce a simple pole at a, so that F satisfies F (s+δ) = F (s)/(s−a), we can use either of, (19)

δ −s/δ B1 (a) ≡ Γ (s − a)/δ

B2 (a) ≡ δ

−s/δ

Γ (a + δ − s)/δ cos(π(s − r)/δ)

Using these elementary terms, we can easily write down solutions to the Bessel and hypergeometric differential equations in the Mellin domain. Since there are two possible elementary terms corresponding to each zero and pole introduced, if the total number of zeros and poles introduced is n, then we can formally write down 2n solutions.

3.3. Bessel’s equation. The Mellin transform of Bessel’s differential equation yields the difference equation, f˜(s + 2) = −(s + ν)(s − ν)f˜(s)

(20)

The elementary terms corresponding to the creation of a zero at −ν and at ν are, s+ν A1 (−ν) = 2s/2 Γ 2 A2 (−ν) =

s−ν A1 (ν) = 2s/2 Γ 2

2s/2 −ν + 2 − s Γ C1 (−ν) 2

A2 (ν) =

C1 (±ν) = cos

π(s ∓ ν) 2

2s/2 ν+2−s Γ C1 (ν) 2

These elements are depicted graphically in the following figure,

···

··· x x x x x ··· ··· x x x x x ··· b bA2b b · · · b bA2b b · · · A1 A1 x x x x ··· x x x x −ν

···

b bC1b b b

0 ···

···

ν

b bC1b b b

···

where the x’s represent poles and the circles represent zeros, and each sequence is on the line Im(s) = 0, though separated vertically in the figure for clarity. Real parts are accurate. We take one of the −ν terms, one of the ν terms, and a cosine term, so that when everything is shifted to the left by two, the effect is to introduce zeros at −ν and ν, and multiply by −1. The cosine terms are also chosen to ensure that the function is integrable on some vertical line Re(s) = k, so that the Mellin transform can be inverted. Using (17) to write the cosine expressions in terms of Γ-functions, we have the following four basic solutions,

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J. A. PALMER

s+ν s−ν Γ Γ 2 2 A1 A1 C1 = π 2s 1+s−r 1−s−r Γ Γ 2 2 s+ν Γ 2 A1 A2 C1 = 2s ν+2−s Γ 2

s−ν Γ 2 A2 A1 C1 = 2s −ν + 2 − s Γ 2

1−s−r 1+s−r Γ Γ 2 2 A2 A2 C1 = s+ν s−ν π Γ Γ 2 2 2s

where r is arbitrary. Each of these expressions is a solution to (20). They will be shown to correspond respectively to Yν (Bessel function of the second kind), Jν , J−ν (Bessel functions of the first kind), and H−ν (Struve’s function of order −ν).

3.3.1. Series solutions. Using the theory of residues and contour integration, together with the asymptotic properties of the Γ-function, we can easily derive series solutions corresponding to each solution to the complex difference equation. For A1 A1 C1 , we have f˜(σ + it) = O(||) According to Theorem 1

3.3.2. Integral representations.

3.4. Hypergeometric equation. We have seen that the Hypergeometric differential equation gives rise, after rearranging (12), to the following quasi-periodic functional equation, (21)

f˜(s + 1) =

s s − (c − 1) f˜(s) s − (a − 1) s − (b − 1)

Here δ is 1, and the elementary terms are, A1 (0) =

Γ(s)

A1 (c − 1) = Γ(s + 1 − c) 1 Γ(s + 1 − a) 1 B1 (b − 1) = Γ(s + 1 − b) B1 (a − 1) =

1 Γ(1 − s)C1 1 A2 (c − 1) = Γ(c − s)C1 A2 (0) =

B2 (a − 1) = Γ(a − s)C1 B2 (b − 1) = Γ(b − s)C1

where C1 indicates some kind of cosine term of the form cos(π(s−r)) or 1/ cos(π(s− r)). The purpose of these terms with respect to the functional is to multiply by −1, so any two of the C1 terms multiplied or divided can be chosen to cancel each other out, so long as the resulting function is invertible. Taking the 16 possible combinations of the elementary terms, and agreeing to cancel cosine terms whenever possible and limit the number of Γ functions in the numerator or denominator to four, we obtain the following basic solutions,

SOLUTION OF DIFFERENTIAL EQUATIONS OF HYPERGEOMETRIC TYPE

A1 A 1 B 1 B1 =

Γ(s)Γ(s+1−c) Γ(s+1−a)Γ(s+1−b)

A2 A 1 B 1 B1 =

A1 A 1 B 1 B2 =

π Γ(s)Γ(s+1−c)Γ(b−s) Γ(s+1−a)Γ(r+s)Γ(1−r−s)

A2 A 1 B 1 B2 =

Γ(s+1−c)Γ(b−s) Γ(1−s)Γ(s+1−a)

A1 A 1 B 2 B1 =

π Γ(s)Γ(s+1−c)Γ(a−s) Γ(s+1−b)Γ(r+s)Γ(1−r−s)

A2 A1 B 2 B1 =

Γ(s+1−c)Γ(a−s) Γ(1−s)Γ(s+1−b)

A1 A1 B2 B2 = Γ(s)Γ(s+1−c)Γ(a−s)Γ(b−s) A 1 A2 B 1 B1 =

Γ(s)Γ(r+s)Γ(1−r−s) π Γ(s+1−a)Γ(s+1−b)Γ(c−s)

A2 A 1 B 2 B2 = A2 A2 B 1 B1 =

A 1 A2 B 1 B2 =

Γ(s)Γ(b−s) Γ(s+1−a)Γ(c−s)

A2 A2 B 1 B2 =

A 1 A2 B 2 B1 =

Γ(s)Γ(a−s) Γ(s+1−b)Γ(c−s)

A2 A 2 B 2 B1 =

A 1 A2 B 2 B2 =

π Γ(s)Γ(a−s)Γ(b−s) Γ(c−s)Γ(r+s)Γ(1−r−s)

A 2 A2 B 2 B2 =

9

Γ(s+1−c)Γ(r+s)Γ(1−r−s) π Γ(1−s)Γ(s+1−a)Γ(s+1−b)

π Γ(s+1−c)Γ(a−s)Γ(b−s)

Γ(1−s)Γ(r+s)Γ(1−r−s) Γ(r+s)Γ(1−r−s)Γ(r 0 +s)Γ(1−r 0 −s) π2 Γ(1−s)Γ(c−s)Γ(s+1−a)Γ(s+1−b)

Γ(r+s)Γ(1−r−s)Γ(b−s) π Γ(1−s)Γ(c−s)Γ(s+1−a)

Γ(r+s)Γ(1−r−s)Γ(a−s) π Γ(1−s)Γ(c−s)Γ(s+1−b)

Γ(a−s)Γ(b−s) Γ(1−s)Γ(c−s)

where r and r0 are arbitrary. 3.5. Series solutions. We now show how to determine solutions in series form using the inverse transform (3) and the method of contour integration. 3.6. Integral solutions. 3.7. Inverse operators. The following are inverse operators. If a ≥ 0, Z ∞ Z x T = a + xD T −1 = x−a ta−1 f (t) dt or x−a ta−1 f (t) dt 0

x

depending on which integral exists. 4. Criteria for hypergeometric type 4.1. Transforming by multiplicative factor. Consider the following second order differential equaiton, p0 (x)y 00 (x) + p1 (x)y 0 (x) + p2 (x)y(x) = 0 Given Define the function f (x) such that y(x) = w(x)f (x). References [1] Lokenath Debnath. Integral transforms and their applications. CRC Press, 1995. [2] Einar Hille. Ordinary differential equations in the complex domain. New York: John Wiley and Sons, 1964. [3] E. L. Ince. Ordinary differential equations. New York: Dover Publications, Inc., 1964. [4] O. I. Marichev. Handbook of integral transforms of higher transcendental functions. New York: Halsted Press, 1964. Translated by L. W. Longdon. [5] I. N. Sneddon. The use of integral transforms. McGraw-Hill, 1972. [6] E. C. Titchmarsh. The theory of functions. Oxford University Press, second edition, 1939. [7] E. C. Titchmarsh. Introduction to the theory of Fourier integrals. Oxford: Clarendon Press, second edition, 1948. [8] E. T. Whittaker and G. N. Watson. A course of modern analysis. Cambridge: The University Press; New York: The Macmillan Company, American edition, 1944.

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J. A. PALMER

Electrical and Computer Engineering Dept., Univ. of California San Diego Current address: La Jolla, CA 92093 E-mail address: [email protected]

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