Mixing
Can Bessel functions be represented as a single function with two variables?
$begingroup$
The typical way to represent a Bessel function of first kind is $ J_{alpha}(z)$, i.e. $ J_{alpha}: mathbb{C}to mathbb{C}$.
Is there any good reason that prevents us to write it as a function of two variables $J(alpha,z)$, i.e. $ J: mathbb{C}^2to mathbb{C}$ ?
complex-analysis ordinary-differential-equations bessel-functions
asked Jan 8 at 12:47
Alessandro ZuninoAlessandro Zunino
15311
$endgroup$
add a comment |
$begingroup$
The typical way to represent a Bessel function of first kind is $ J_{alpha}(z)$, i.e. $ J_{alpha}: mathbb{C}to mathbb{C}$.
Is there any good reason that prevents us to write it as a function of two variables $J(alpha,z)$, i.e. $ J: mathbb{C}^2to mathbb{C}$ ?
complex-analysis ordinary-differential-equations bessel-functions
asked Jan 8 at 12:47
Alessandro ZuninoAlessandro Zunino
15311
$endgroup$
$begingroup$
The Maple version looks like this:BesselJ(a, z).
$endgroup$
– GEdgar
Jan 8 at 13:55
add a comment |
$begingroup$
The typical way to represent a Bessel function of first kind is $ J_{alpha}(z)$, i.e. $ J_{alpha}: mathbb{C}to mathbb{C}$.
Is there any good reason that prevents us to write it as a function of two variables $J(alpha,z)$, i.e. $ J: mathbb{C}^2to mathbb{C}$ ?
complex-analysis ordinary-differential-equations bessel-functions
asked Jan 8 at 12:47
Alessandro ZuninoAlessandro Zunino
15311
$endgroup$
The typical way to represent a Bessel function of first kind is $ J_{alpha}(z)$, i.e. $ J_{alpha}: mathbb{C}to mathbb{C}$.
Is there any good reason that prevents us to write it as a function of two variables $J(alpha,z)$, i.e. $ J: mathbb{C}^2to mathbb{C}$ ?
complex-analysis ordinary-differential-equations bessel-functions
complex-analysis ordinary-differential-equations bessel-functions
asked Jan 8 at 12:47
Alessandro ZuninoAlessandro Zunino
15311
asked Jan 8 at 12:47
Alessandro ZuninoAlessandro Zunino
15311
edited Jan 8 at 18:19
Alessandro Zunino
asked Jan 8 at 12:47
Alessandro ZuninoAlessandro Zunino
15311
asked Jan 8 at 12:47
Alessandro ZuninoAlessandro Zunino
15311
asked Jan 8 at 12:47
Alessandro ZuninoAlessandro Zunino
15311
15311
$begingroup$
The Maple version looks like this:BesselJ(a, z).
$endgroup$
– GEdgar
Jan 8 at 13:55
add a comment |
$begingroup$
The Maple version looks like this:BesselJ(a, z).
$endgroup$
– GEdgar
Jan 8 at 13:55
$begingroup$
The Maple version looks like this:
BesselJ(a, z).$endgroup$
– GEdgar
Jan 8 at 13:55
$begingroup$
The Maple version looks like this:
BesselJ(a, z).$endgroup$
– GEdgar
Jan 8 at 13:55
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
No, there is nothing to prevent considering $J_alpha(z)$ as a function of both $alpha$ and $z$. However, not quite on $mathbb C^2$: it has a branch point at $z=0$ if $alpha$ is not an integer.
answered Jan 8 at 13:01
Robert IsraelRobert Israel
321k23210463
$endgroup$
$begingroup$
So, if I always see it written as a function of just one variable is just because of convention?
$endgroup$
– Alessandro Zunino
Jan 8 at 13:03
$begingroup$
@AlessandroZunino : It is also easier to write the defining ODE for a function of a single variable with a parameter. In the two-variable form it would be a PDE where the $z$ derivatives are now partial derivatives.
$endgroup$
– LutzL
Jan 8 at 14:44
$begingroup$
@LutzL Turning the Bessel differential equations from ODE to PDE does change the solutions or the way to obtain them? In theory no, so why should be easier to consider $alpha$ a fixed parameter?
$endgroup$
– Alessandro Zunino
Jan 8 at 17:32
$begingroup$
@AlessandroZunino : It makes the topic look more complicated, as PDE theory in general is more complex than ODE theory, and the question is about optics, of mixing old-style and modern-style variable positioning. $α$ is a fixed parameter in the Bessel equation, I'm not sure what that has to do with this styling problem.
$endgroup$
– LutzL
Jan 8 at 17:47
2
$begingroup$
It's not "really" a PDE, since the derivatives are only with respect to $z$, not $alpha$. It's still just an ODE with a parameter.
$endgroup$
– Robert Israel
Jan 8 at 19:00
add a comment |
$begingroup$
It depends very much on the context:
In physics, Bessel functions often appear in problems with cylindrical or spherical symmetry where $z$ plays the role of appropriate radial coordinate. The parameter $alpha$ is usually fixed by some quantization condition but is not directly related to physical coordinates.
In the theory of linear ODEs with rational coefficients, the Bessel equation corresponds to a model rank 2 system with prescribed structure of singularities on the Riemann sphere $mathbb Cmathbb P^1$ (one regular and one ramified irregular singularity of Poincaré rank 1). Here $z$ corresponds to a canonical coordinate on $mathbb Cmathbb P^1$ whereas $alpha$ encodes monodromy properties of the system. So again, the nature of the two parameters is rather different.
On the other hand, in addition to 2nd order differential equation w.r.t. $z$, Bessel function also satisfies 2nd order difference equation (recurrence relation) w.r.t. $alpha$. It is possible to think of these equations in the same way. However, in order to see the analogy between $z$ and $alpha$ more clearly, one needs to discretize the first parameter and go into the world of $q$-special functions.
answered Jan 8 at 15:59
Start wearing purpleStart wearing purple
47.2k12135192
$endgroup$
$begingroup$
The context, in my case, is Physics. The problem I am facing makes use of Bessel function in which both $z$ and $alpha$ have a deep physical meaning and they are both variable (just $alpha$ is constrained to be an integer). Can you please explain better your last point? I am very interested in understanding this analogy between $z$ and $alpha$ (I admit I don't know a lot about $q$-special functions).
$endgroup$
– Alessandro Zunino
Jan 8 at 18:02
1
$begingroup$
@AlessandroZunino The recurrence relation for Bessel function involves $J_alpha$ ,$J_{alpha+1}$ and $J_{alpha-1}$. Imagine that $alpha$ is very large so that its shifts by $pm1$ can be considered as small. One could also imagine that after appropriate scaling, the recurrence relation could then become a 2nd order differential equation. Now what one should do is actually the inverse of this procedure: guess a recurrence relation in the appropriately rescaled variable $z$ which gives the Bessel equation in the limit.
$endgroup$
– Start wearing purple
Jan 8 at 21:59
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No, there is nothing to prevent considering $J_alpha(z)$ as a function of both $alpha$ and $z$. However, not quite on $mathbb C^2$: it has a branch point at $z=0$ if $alpha$ is not an integer.
answered Jan 8 at 13:01
Robert IsraelRobert Israel
321k23210463
$endgroup$
$begingroup$
So, if I always see it written as a function of just one variable is just because of convention?
$endgroup$
– Alessandro Zunino
Jan 8 at 13:03
$begingroup$
@AlessandroZunino : It is also easier to write the defining ODE for a function of a single variable with a parameter. In the two-variable form it would be a PDE where the $z$ derivatives are now partial derivatives.
$endgroup$
– LutzL
Jan 8 at 14:44
$begingroup$
@LutzL Turning the Bessel differential equations from ODE to PDE does change the solutions or the way to obtain them? In theory no, so why should be easier to consider $alpha$ a fixed parameter?
$endgroup$
– Alessandro Zunino
Jan 8 at 17:32
$begingroup$
@AlessandroZunino : It makes the topic look more complicated, as PDE theory in general is more complex than ODE theory, and the question is about optics, of mixing old-style and modern-style variable positioning. $α$ is a fixed parameter in the Bessel equation, I'm not sure what that has to do with this styling problem.
$endgroup$
– LutzL
Jan 8 at 17:47
2
$begingroup$
It's not "really" a PDE, since the derivatives are only with respect to $z$, not $alpha$. It's still just an ODE with a parameter.
$endgroup$
– Robert Israel
Jan 8 at 19:00
add a comment |
$begingroup$
No, there is nothing to prevent considering $J_alpha(z)$ as a function of both $alpha$ and $z$. However, not quite on $mathbb C^2$: it has a branch point at $z=0$ if $alpha$ is not an integer.
answered Jan 8 at 13:01
Robert IsraelRobert Israel
321k23210463
$endgroup$
$begingroup$
So, if I always see it written as a function of just one variable is just because of convention?
$endgroup$
– Alessandro Zunino
Jan 8 at 13:03
$begingroup$
@AlessandroZunino : It is also easier to write the defining ODE for a function of a single variable with a parameter. In the two-variable form it would be a PDE where the $z$ derivatives are now partial derivatives.
$endgroup$
– LutzL
Jan 8 at 14:44
$begingroup$
@LutzL Turning the Bessel differential equations from ODE to PDE does change the solutions or the way to obtain them? In theory no, so why should be easier to consider $alpha$ a fixed parameter?
$endgroup$
– Alessandro Zunino
Jan 8 at 17:32
$begingroup$
@AlessandroZunino : It makes the topic look more complicated, as PDE theory in general is more complex than ODE theory, and the question is about optics, of mixing old-style and modern-style variable positioning. $α$ is a fixed parameter in the Bessel equation, I'm not sure what that has to do with this styling problem.
$endgroup$
– LutzL
Jan 8 at 17:47
2
$begingroup$
It's not "really" a PDE, since the derivatives are only with respect to $z$, not $alpha$. It's still just an ODE with a parameter.
$endgroup$
– Robert Israel
Jan 8 at 19:00
add a comment |
$begingroup$
No, there is nothing to prevent considering $J_alpha(z)$ as a function of both $alpha$ and $z$. However, not quite on $mathbb C^2$: it has a branch point at $z=0$ if $alpha$ is not an integer.
answered Jan 8 at 13:01
Robert IsraelRobert Israel
321k23210463
$endgroup$
No, there is nothing to prevent considering $J_alpha(z)$ as a function of both $alpha$ and $z$. However, not quite on $mathbb C^2$: it has a branch point at $z=0$ if $alpha$ is not an integer.
answered Jan 8 at 13:01
Robert IsraelRobert Israel
321k23210463
answered Jan 8 at 13:01
Robert IsraelRobert Israel
321k23210463
answered Jan 8 at 13:01
Robert IsraelRobert Israel
321k23210463
answered Jan 8 at 13:01
Robert IsraelRobert Israel
321k23210463
321k23210463
$begingroup$
So, if I always see it written as a function of just one variable is just because of convention?
$endgroup$
– Alessandro Zunino
Jan 8 at 13:03
$begingroup$
@AlessandroZunino : It is also easier to write the defining ODE for a function of a single variable with a parameter. In the two-variable form it would be a PDE where the $z$ derivatives are now partial derivatives.
$endgroup$
– LutzL
Jan 8 at 14:44
$begingroup$
@LutzL Turning the Bessel differential equations from ODE to PDE does change the solutions or the way to obtain them? In theory no, so why should be easier to consider $alpha$ a fixed parameter?
$endgroup$
– Alessandro Zunino
Jan 8 at 17:32
$begingroup$
@AlessandroZunino : It makes the topic look more complicated, as PDE theory in general is more complex than ODE theory, and the question is about optics, of mixing old-style and modern-style variable positioning. $α$ is a fixed parameter in the Bessel equation, I'm not sure what that has to do with this styling problem.
$endgroup$
– LutzL
Jan 8 at 17:47
2
$begingroup$
It's not "really" a PDE, since the derivatives are only with respect to $z$, not $alpha$. It's still just an ODE with a parameter.
$endgroup$
– Robert Israel
Jan 8 at 19:00
add a comment |
$begingroup$
So, if I always see it written as a function of just one variable is just because of convention?
$endgroup$
– Alessandro Zunino
Jan 8 at 13:03
$begingroup$
@AlessandroZunino : It is also easier to write the defining ODE for a function of a single variable with a parameter. In the two-variable form it would be a PDE where the $z$ derivatives are now partial derivatives.
$endgroup$
– LutzL
Jan 8 at 14:44
$begingroup$
@LutzL Turning the Bessel differential equations from ODE to PDE does change the solutions or the way to obtain them? In theory no, so why should be easier to consider $alpha$ a fixed parameter?
$endgroup$
– Alessandro Zunino
Jan 8 at 17:32
$begingroup$
@AlessandroZunino : It makes the topic look more complicated, as PDE theory in general is more complex than ODE theory, and the question is about optics, of mixing old-style and modern-style variable positioning. $α$ is a fixed parameter in the Bessel equation, I'm not sure what that has to do with this styling problem.
$endgroup$
– LutzL
Jan 8 at 17:47
2
$begingroup$
It's not "really" a PDE, since the derivatives are only with respect to $z$, not $alpha$. It's still just an ODE with a parameter.
$endgroup$
– Robert Israel
Jan 8 at 19:00
$begingroup$
So, if I always see it written as a function of just one variable is just because of convention?
$endgroup$
– Alessandro Zunino
Jan 8 at 13:03
$begingroup$
So, if I always see it written as a function of just one variable is just because of convention?
$endgroup$
– Alessandro Zunino
Jan 8 at 13:03
$begingroup$
@AlessandroZunino : It is also easier to write the defining ODE for a function of a single variable with a parameter. In the two-variable form it would be a PDE where the $z$ derivatives are now partial derivatives.
$endgroup$
– LutzL
Jan 8 at 14:44
$begingroup$
@AlessandroZunino : It is also easier to write the defining ODE for a function of a single variable with a parameter. In the two-variable form it would be a PDE where the $z$ derivatives are now partial derivatives.
$endgroup$
– LutzL
Jan 8 at 14:44
$begingroup$
@LutzL Turning the Bessel differential equations from ODE to PDE does change the solutions or the way to obtain them? In theory no, so why should be easier to consider $alpha$ a fixed parameter?
$endgroup$
– Alessandro Zunino
Jan 8 at 17:32
$begingroup$
@LutzL Turning the Bessel differential equations from ODE to PDE does change the solutions or the way to obtain them? In theory no, so why should be easier to consider $alpha$ a fixed parameter?
$endgroup$
– Alessandro Zunino
Jan 8 at 17:32
$begingroup$
@AlessandroZunino : It makes the topic look more complicated, as PDE theory in general is more complex than ODE theory, and the question is about optics, of mixing old-style and modern-style variable positioning. $α$ is a fixed parameter in the Bessel equation, I'm not sure what that has to do with this styling problem.
$endgroup$
– LutzL
Jan 8 at 17:47
$begingroup$
@AlessandroZunino : It makes the topic look more complicated, as PDE theory in general is more complex than ODE theory, and the question is about optics, of mixing old-style and modern-style variable positioning. $α$ is a fixed parameter in the Bessel equation, I'm not sure what that has to do with this styling problem.
$endgroup$
– LutzL
Jan 8 at 17:47
2
2
$begingroup$
It's not "really" a PDE, since the derivatives are only with respect to $z$, not $alpha$. It's still just an ODE with a parameter.
$endgroup$
– Robert Israel
Jan 8 at 19:00
$begingroup$
It's not "really" a PDE, since the derivatives are only with respect to $z$, not $alpha$. It's still just an ODE with a parameter.
$endgroup$
– Robert Israel
Jan 8 at 19:00
add a comment |
$begingroup$
It depends very much on the context:
In physics, Bessel functions often appear in problems with cylindrical or spherical symmetry where $z$ plays the role of appropriate radial coordinate. The parameter $alpha$ is usually fixed by some quantization condition but is not directly related to physical coordinates.
In the theory of linear ODEs with rational coefficients, the Bessel equation corresponds to a model rank 2 system with prescribed structure of singularities on the Riemann sphere $mathbb Cmathbb P^1$ (one regular and one ramified irregular singularity of Poincaré rank 1). Here $z$ corresponds to a canonical coordinate on $mathbb Cmathbb P^1$ whereas $alpha$ encodes monodromy properties of the system. So again, the nature of the two parameters is rather different.
On the other hand, in addition to 2nd order differential equation w.r.t. $z$, Bessel function also satisfies 2nd order difference equation (recurrence relation) w.r.t. $alpha$. It is possible to think of these equations in the same way. However, in order to see the analogy between $z$ and $alpha$ more clearly, one needs to discretize the first parameter and go into the world of $q$-special functions.
answered Jan 8 at 15:59
Start wearing purpleStart wearing purple
47.2k12135192
$endgroup$
$begingroup$
The context, in my case, is Physics. The problem I am facing makes use of Bessel function in which both $z$ and $alpha$ have a deep physical meaning and they are both variable (just $alpha$ is constrained to be an integer). Can you please explain better your last point? I am very interested in understanding this analogy between $z$ and $alpha$ (I admit I don't know a lot about $q$-special functions).
$endgroup$
– Alessandro Zunino
Jan 8 at 18:02
1
$begingroup$
@AlessandroZunino The recurrence relation for Bessel function involves $J_alpha$ ,$J_{alpha+1}$ and $J_{alpha-1}$. Imagine that $alpha$ is very large so that its shifts by $pm1$ can be considered as small. One could also imagine that after appropriate scaling, the recurrence relation could then become a 2nd order differential equation. Now what one should do is actually the inverse of this procedure: guess a recurrence relation in the appropriately rescaled variable $z$ which gives the Bessel equation in the limit.
$endgroup$
– Start wearing purple
Jan 8 at 21:59
add a comment |
$begingroup$
It depends very much on the context:
In physics, Bessel functions often appear in problems with cylindrical or spherical symmetry where $z$ plays the role of appropriate radial coordinate. The parameter $alpha$ is usually fixed by some quantization condition but is not directly related to physical coordinates.
In the theory of linear ODEs with rational coefficients, the Bessel equation corresponds to a model rank 2 system with prescribed structure of singularities on the Riemann sphere $mathbb Cmathbb P^1$ (one regular and one ramified irregular singularity of Poincaré rank 1). Here $z$ corresponds to a canonical coordinate on $mathbb Cmathbb P^1$ whereas $alpha$ encodes monodromy properties of the system. So again, the nature of the two parameters is rather different.
On the other hand, in addition to 2nd order differential equation w.r.t. $z$, Bessel function also satisfies 2nd order difference equation (recurrence relation) w.r.t. $alpha$. It is possible to think of these equations in the same way. However, in order to see the analogy between $z$ and $alpha$ more clearly, one needs to discretize the first parameter and go into the world of $q$-special functions.
answered Jan 8 at 15:59
Start wearing purpleStart wearing purple
47.2k12135192
$endgroup$
$begingroup$
The context, in my case, is Physics. The problem I am facing makes use of Bessel function in which both $z$ and $alpha$ have a deep physical meaning and they are both variable (just $alpha$ is constrained to be an integer). Can you please explain better your last point? I am very interested in understanding this analogy between $z$ and $alpha$ (I admit I don't know a lot about $q$-special functions).
$endgroup$
– Alessandro Zunino
Jan 8 at 18:02
1
$begingroup$
@AlessandroZunino The recurrence relation for Bessel function involves $J_alpha$ ,$J_{alpha+1}$ and $J_{alpha-1}$. Imagine that $alpha$ is very large so that its shifts by $pm1$ can be considered as small. One could also imagine that after appropriate scaling, the recurrence relation could then become a 2nd order differential equation. Now what one should do is actually the inverse of this procedure: guess a recurrence relation in the appropriately rescaled variable $z$ which gives the Bessel equation in the limit.
$endgroup$
– Start wearing purple
Jan 8 at 21:59
add a comment |
$begingroup$
It depends very much on the context:
In physics, Bessel functions often appear in problems with cylindrical or spherical symmetry where $z$ plays the role of appropriate radial coordinate. The parameter $alpha$ is usually fixed by some quantization condition but is not directly related to physical coordinates.
In the theory of linear ODEs with rational coefficients, the Bessel equation corresponds to a model rank 2 system with prescribed structure of singularities on the Riemann sphere $mathbb Cmathbb P^1$ (one regular and one ramified irregular singularity of Poincaré rank 1). Here $z$ corresponds to a canonical coordinate on $mathbb Cmathbb P^1$ whereas $alpha$ encodes monodromy properties of the system. So again, the nature of the two parameters is rather different.
On the other hand, in addition to 2nd order differential equation w.r.t. $z$, Bessel function also satisfies 2nd order difference equation (recurrence relation) w.r.t. $alpha$. It is possible to think of these equations in the same way. However, in order to see the analogy between $z$ and $alpha$ more clearly, one needs to discretize the first parameter and go into the world of $q$-special functions.
answered Jan 8 at 15:59
Start wearing purpleStart wearing purple
47.2k12135192
$endgroup$
It depends very much on the context:
In physics, Bessel functions often appear in problems with cylindrical or spherical symmetry where $z$ plays the role of appropriate radial coordinate. The parameter $alpha$ is usually fixed by some quantization condition but is not directly related to physical coordinates.
In the theory of linear ODEs with rational coefficients, the Bessel equation corresponds to a model rank 2 system with prescribed structure of singularities on the Riemann sphere $mathbb Cmathbb P^1$ (one regular and one ramified irregular singularity of Poincaré rank 1). Here $z$ corresponds to a canonical coordinate on $mathbb Cmathbb P^1$ whereas $alpha$ encodes monodromy properties of the system. So again, the nature of the two parameters is rather different.
On the other hand, in addition to 2nd order differential equation w.r.t. $z$, Bessel function also satisfies 2nd order difference equation (recurrence relation) w.r.t. $alpha$. It is possible to think of these equations in the same way. However, in order to see the analogy between $z$ and $alpha$ more clearly, one needs to discretize the first parameter and go into the world of $q$-special functions.
answered Jan 8 at 15:59
Start wearing purpleStart wearing purple
47.2k12135192
answered Jan 8 at 15:59
Start wearing purpleStart wearing purple
47.2k12135192
answered Jan 8 at 15:59
Start wearing purpleStart wearing purple
47.2k12135192
answered Jan 8 at 15:59
Start wearing purpleStart wearing purple
47.2k12135192
47.2k12135192
$begingroup$
The context, in my case, is Physics. The problem I am facing makes use of Bessel function in which both $z$ and $alpha$ have a deep physical meaning and they are both variable (just $alpha$ is constrained to be an integer). Can you please explain better your last point? I am very interested in understanding this analogy between $z$ and $alpha$ (I admit I don't know a lot about $q$-special functions).
$endgroup$
– Alessandro Zunino
Jan 8 at 18:02
1
$begingroup$
@AlessandroZunino The recurrence relation for Bessel function involves $J_alpha$ ,$J_{alpha+1}$ and $J_{alpha-1}$. Imagine that $alpha$ is very large so that its shifts by $pm1$ can be considered as small. One could also imagine that after appropriate scaling, the recurrence relation could then become a 2nd order differential equation. Now what one should do is actually the inverse of this procedure: guess a recurrence relation in the appropriately rescaled variable $z$ which gives the Bessel equation in the limit.
$endgroup$
– Start wearing purple
Jan 8 at 21:59
add a comment |
$begingroup$
The context, in my case, is Physics. The problem I am facing makes use of Bessel function in which both $z$ and $alpha$ have a deep physical meaning and they are both variable (just $alpha$ is constrained to be an integer). Can you please explain better your last point? I am very interested in understanding this analogy between $z$ and $alpha$ (I admit I don't know a lot about $q$-special functions).
$endgroup$
– Alessandro Zunino
Jan 8 at 18:02
1
$begingroup$
@AlessandroZunino The recurrence relation for Bessel function involves $J_alpha$ ,$J_{alpha+1}$ and $J_{alpha-1}$. Imagine that $alpha$ is very large so that its shifts by $pm1$ can be considered as small. One could also imagine that after appropriate scaling, the recurrence relation could then become a 2nd order differential equation. Now what one should do is actually the inverse of this procedure: guess a recurrence relation in the appropriately rescaled variable $z$ which gives the Bessel equation in the limit.
$endgroup$
– Start wearing purple
Jan 8 at 21:59
$begingroup$
The context, in my case, is Physics. The problem I am facing makes use of Bessel function in which both $z$ and $alpha$ have a deep physical meaning and they are both variable (just $alpha$ is constrained to be an integer). Can you please explain better your last point? I am very interested in understanding this analogy between $z$ and $alpha$ (I admit I don't know a lot about $q$-special functions).
$endgroup$
– Alessandro Zunino
Jan 8 at 18:02
$begingroup$
The context, in my case, is Physics. The problem I am facing makes use of Bessel function in which both $z$ and $alpha$ have a deep physical meaning and they are both variable (just $alpha$ is constrained to be an integer). Can you please explain better your last point? I am very interested in understanding this analogy between $z$ and $alpha$ (I admit I don't know a lot about $q$-special functions).
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– Alessandro Zunino
Jan 8 at 18:02
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@AlessandroZunino The recurrence relation for Bessel function involves $J_alpha$ ,$J_{alpha+1}$ and $J_{alpha-1}$. Imagine that $alpha$ is very large so that its shifts by $pm1$ can be considered as small. One could also imagine that after appropriate scaling, the recurrence relation could then become a 2nd order differential equation. Now what one should do is actually the inverse of this procedure: guess a recurrence relation in the appropriately rescaled variable $z$ which gives the Bessel equation in the limit.
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– Start wearing purple
Jan 8 at 21:59
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@AlessandroZunino The recurrence relation for Bessel function involves $J_alpha$ ,$J_{alpha+1}$ and $J_{alpha-1}$. Imagine that $alpha$ is very large so that its shifts by $pm1$ can be considered as small. One could also imagine that after appropriate scaling, the recurrence relation could then become a 2nd order differential equation. Now what one should do is actually the inverse of this procedure: guess a recurrence relation in the appropriately rescaled variable $z$ which gives the Bessel equation in the limit.
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– Start wearing purple
Jan 8 at 21:59
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The Maple version looks like this:
BesselJ(a, z).$endgroup$
– GEdgar
Jan 8 at 13:55