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gfun

  

`diffeq+diffeq`

  

determine the differential equation satisfied by the sum of two holonomic functions

  

`diffeq*diffeq`

  

determine the differential equation satisfied by the Cauchy product of two holonomic functions

  

hadamardproduct

  

determine the differential equation satisfied by the Hadamard product of two holonomic functions

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

`diffeq+diffeq`(eq1, eq2, y(z))

`diffeq*diffeq`(eq1, eq2, y(z))

hadamardproduct(eq1, eq2, y(z))

Parameters

eq1, eq2

-

two linear differential equations with polynomial coefficients

y

-

name; holonomic function name

z

-

name; variable of the holonomic function y

Description

• 

The gfun[`diffeq+diffeq`](eq1, eq2, y(z)) command determines the differential equation satisfied by the sum of two holonomic functions, eq1 and eq2.

  

If f and g are holonomic function solutions of eq1 and eq2, respectively, the gfun[`diffeq+diffeq`] function returns a linear differential equation verified by f+g.

  

The differential order of the returned equation is at most the sum of eq1 and eq2's differential orders for the gfun[`diffeq+diffeq`] function.

• 

The gfun[`diffeq*diffeq`](eq1, eq2, y(z)) command determines the differential equation satisfied by the Cauchy product of two holonomic functions, eq1 and eq2.

  

If f and g are holonomic function solutions of eq1 and eq2), respectively, the gfun[`diffeq*diffeq`] function returns a linear differential equation verified by fg.

  

The differential order of the returned equation is at most the product of eq1 and eq2's differential orders for the gfun[`diffeq*diffeq`] function.

• 

The gfun[hadamardproduct](eq1, eq2, y(z)) command determines the differential equation satisfied by the Hadamard product of two holonomic functions, eq1 and eq2.

  

If f and g are holonomic function solutions of eq1 and eq2, respectively, the gfun[hadamardproduct] function returns a linear differential equation verified by the Hadamard product of f and g.  The function whose coefficient of zn in the Taylor expansion around 0 is the product of the corresponding coefficients of f and g.

Examples

withgfun:

eq1Dyxyx:

eq21+xD2yx+Dyx:

`diffeq+diffeq`eq1,eq2,yx

x3ⅆⅆxyx+x22x+1ⅆ2ⅆx2yx+x2+3x+2ⅆ3ⅆx3yx

(1)

`diffeq*diffeq`eq1,eq2,yx

xyx+2x1ⅆⅆxyx+1+xⅆ2ⅆx2yx

(2)

hadamardproducteq1,eq2,yx

1+xⅆⅆxyx+ⅆ2ⅆx2yxx_t2_C1,y0=_t2_C0

(3)

See Also

gfun

gfun[`rec+rec`]

gfun[`rec*rec`]

gfun[cauchyproduct]

gfun[poltodiffeq]