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backend_service/venv/lib/python3.13/site-packages/sympy/polys/appellseqs.py

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+r"""
+Efficient functions for generating Appell sequences.
+
+An Appell sequence is a zero-indexed sequence of polynomials `p_i(x)`
+satisfying `p_{i+1}'(x)=(i+1)p_i(x)` for all `i`. This definition leads
+to the following iterative algorithm:
+
+.. math :: p_0(x) = c_0,\ p_i(x) = i \int_0^x p_{i-1}(t)\,dt + c_i
+
+The constant coefficients `c_i` are usually determined from the
+just-evaluated integral and `i`.
+
+Appell sequences satisfy the following identity from umbral calculus:
+
+.. math :: p_n(x+y) = \sum_{k=0}^n \binom{n}{k} p_k(x) y^{n-k}
+
+References
+==========
+
+.. [1] https://en.wikipedia.org/wiki/Appell_sequence
+.. [2] Peter Luschny, "An introduction to the Bernoulli function",
+       https://arxiv.org/abs/2009.06743
+"""
+from sympy.polys.densearith import dup_mul_ground, dup_sub_ground, dup_quo_ground
+from sympy.polys.densetools import dup_eval, dup_integrate
+from sympy.polys.domains import ZZ, QQ
+from sympy.polys.polytools import named_poly
+from sympy.utilities import public
+
+def dup_bernoulli(n, K):
+    """Low-level implementation of Bernoulli polynomials."""
+    if n < 1:
+        return [K.one]
+    p = [K.one, K(-1,2)]
+    for i in range(2, n+1):
+        p = dup_integrate(dup_mul_ground(p, K(i), K), 1, K)
+        if i % 2 == 0:
+            p = dup_sub_ground(p, dup_eval(p, K(1,2), K) * K(1<<(i-1), (1<<i)-1), K)
+    return p
+
+@public
+def bernoulli_poly(n, x=None, polys=False):
+    r"""Generates the Bernoulli polynomial `\operatorname{B}_n(x)`.
+
+    `\operatorname{B}_n(x)` is the unique polynomial satisfying
+
+    .. math :: \int_{x}^{x+1} \operatorname{B}_n(t) \,dt = x^n.
+
+    Based on this, we have for nonnegative integer `s` and integer
+    `a` and `b`
+
+    .. math :: \sum_{k=a}^{b} k^s = \frac{\operatorname{B}_{s+1}(b+1) -
+            \operatorname{B}_{s+1}(a)}{s+1}
+
+    which is related to Jakob Bernoulli's original motivation for introducing
+    the Bernoulli numbers, the values of these polynomials at `x = 1`.
+
+    Examples
+    ========
+
+    >>> from sympy import summation
+    >>> from sympy.abc import x
+    >>> from sympy.polys import bernoulli_poly
+    >>> bernoulli_poly(5, x)
+    x**5 - 5*x**4/2 + 5*x**3/3 - x/6
+
+    >>> def psum(p, a, b):
+    ...     return (bernoulli_poly(p+1,b+1) - bernoulli_poly(p+1,a)) / (p+1)
+    >>> psum(4, -6, 27)
+    3144337
+    >>> summation(x**4, (x, -6, 27))
+    3144337
+
+    >>> psum(1, 1, x).factor()
+    x*(x + 1)/2
+    >>> psum(2, 1, x).factor()
+    x*(x + 1)*(2*x + 1)/6
+    >>> psum(3, 1, x).factor()
+    x**2*(x + 1)**2/4
+
+    Parameters
+    ==========
+
+    n : int
+        Degree of the polynomial.
+    x : optional
+    polys : bool, optional
+        If True, return a Poly, otherwise (default) return an expression.
+
+    See Also
+    ========
+
+    sympy.functions.combinatorial.numbers.bernoulli
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Bernoulli_polynomials
+    """
+    return named_poly(n, dup_bernoulli, QQ, "Bernoulli polynomial", (x,), polys)
+
+def dup_bernoulli_c(n, K):
+    """Low-level implementation of central Bernoulli polynomials."""
+    p = [K.one]
+    for i in range(1, n+1):
+        p = dup_integrate(dup_mul_ground(p, K(i), K), 1, K)
+        if i % 2 == 0:
+            p = dup_sub_ground(p, dup_eval(p, K.one, K) * K((1<<(i-1))-1, (1<<i)-1), K)
+    return p
+
+@public
+def bernoulli_c_poly(n, x=None, polys=False):
+    r"""Generates the central Bernoulli polynomial `\operatorname{B}_n^c(x)`.
+
+    These are scaled and shifted versions of the plain Bernoulli polynomials,
+    done in such a way that `\operatorname{B}_n^c(x)` is an even or odd function
+    for even or odd `n` respectively:
+
+    .. math :: \operatorname{B}_n^c(x) = 2^n \operatorname{B}_n
+            \left(\frac{x+1}{2}\right)
+
+    Parameters
+    ==========
+
+    n : int
+        Degree of the polynomial.
+    x : optional
+    polys : bool, optional
+        If True, return a Poly, otherwise (default) return an expression.
+    """
+    return named_poly(n, dup_bernoulli_c, QQ, "central Bernoulli polynomial", (x,), polys)
+
+def dup_genocchi(n, K):
+    """Low-level implementation of Genocchi polynomials."""
+    if n < 1:
+        return [K.zero]
+    p = [-K.one]
+    for i in range(2, n+1):
+        p = dup_integrate(dup_mul_ground(p, K(i), K), 1, K)
+        if i % 2 == 0:
+            p = dup_sub_ground(p, dup_eval(p, K.one, K) // K(2), K)
+    return p
+
+@public
+def genocchi_poly(n, x=None, polys=False):
+    r"""Generates the Genocchi polynomial `\operatorname{G}_n(x)`.
+
+    `\operatorname{G}_n(x)` is twice the difference between the plain and
+    central Bernoulli polynomials, so has degree `n-1`:
+
+    .. math :: \operatorname{G}_n(x) = 2 (\operatorname{B}_n(x) -
+            \operatorname{B}_n^c(x))
+
+    The factor of 2 in the definition endows `\operatorname{G}_n(x)` with
+    integer coefficients.
+
+    Parameters
+    ==========
+
+    n : int
+        Degree of the polynomial plus one.
+    x : optional
+    polys : bool, optional
+        If True, return a Poly, otherwise (default) return an expression.
+
+    See Also
+    ========
+
+    sympy.functions.combinatorial.numbers.genocchi
+    """
+    return named_poly(n, dup_genocchi, ZZ, "Genocchi polynomial", (x,), polys)
+
+def dup_euler(n, K):
+    """Low-level implementation of Euler polynomials."""
+    return dup_quo_ground(dup_genocchi(n+1, ZZ), K(-n-1), K)
+
+@public
+def euler_poly(n, x=None, polys=False):
+    r"""Generates the Euler polynomial `\operatorname{E}_n(x)`.
+
+    These are scaled and reindexed versions of the Genocchi polynomials:
+
+    .. math :: \operatorname{E}_n(x) = -\frac{\operatorname{G}_{n+1}(x)}{n+1}
+
+    Parameters
+    ==========
+
+    n : int
+        Degree of the polynomial.
+    x : optional
+    polys : bool, optional
+        If True, return a Poly, otherwise (default) return an expression.
+
+    See Also
+    ========
+
+    sympy.functions.combinatorial.numbers.euler
+    """
+    return named_poly(n, dup_euler, QQ, "Euler polynomial", (x,), polys)
+
+def dup_andre(n, K):
+    """Low-level implementation of Andre polynomials."""
+    p = [K.one]
+    for i in range(1, n+1):
+        p = dup_integrate(dup_mul_ground(p, K(i), K), 1, K)
+        if i % 2 == 0:
+            p = dup_sub_ground(p, dup_eval(p, K.one, K), K)
+    return p
+
+@public
+def andre_poly(n, x=None, polys=False):
+    r"""Generates the Andre polynomial `\mathcal{A}_n(x)`.
+
+    This is the Appell sequence where the constant coefficients form the sequence
+    of Euler numbers ``euler(n)``. As such they have integer coefficients
+    and parities matching the parity of `n`.
+
+    Luschny calls these the *Swiss-knife polynomials* because their values
+    at 0 and 1 can be simply transformed into both the Bernoulli and Euler
+    numbers. Here they are called the Andre polynomials because
+    `|\mathcal{A}_n(n\bmod 2)|` for `n \ge 0` generates what Luschny calls
+    the *Andre numbers*, A000111 in the OEIS.
+
+    Examples
+    ========
+
+    >>> from sympy import bernoulli, euler, genocchi
+    >>> from sympy.abc import x
+    >>> from sympy.polys import andre_poly
+    >>> andre_poly(9, x)
+    x**9 - 36*x**7 + 630*x**5 - 5124*x**3 + 12465*x
+
+    >>> [andre_poly(n, 0) for n in range(11)]
+    [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521]
+    >>> [euler(n) for n in range(11)]
+    [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521]
+    >>> [andre_poly(n-1, 1) * n / (4**n - 2**n) for n in range(1, 11)]
+    [1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]
+    >>> [bernoulli(n) for n in range(1, 11)]
+    [1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]
+    >>> [-andre_poly(n-1, -1) * n / (-2)**(n-1) for n in range(1, 11)]
+    [-1, -1, 0, 1, 0, -3, 0, 17, 0, -155]
+    >>> [genocchi(n) for n in range(1, 11)]
+    [-1, -1, 0, 1, 0, -3, 0, 17, 0, -155]
+
+    >>> [abs(andre_poly(n, n%2)) for n in range(11)]
+    [1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521]
+
+    Parameters
+    ==========
+
+    n : int
+        Degree of the polynomial.
+    x : optional
+    polys : bool, optional
+        If True, return a Poly, otherwise (default) return an expression.
+
+    See Also
+    ========
+
+    sympy.functions.combinatorial.numbers.andre
+
+    References
+    ==========
+
+    .. [1] Peter Luschny, "An introduction to the Bernoulli function",
+           https://arxiv.org/abs/2009.06743
+    """
+    return named_poly(n, dup_andre, ZZ, "Andre polynomial", (x,), polys)