### 28.1 Functions and Variables for Sums and Products

Function: bashindices (expr)

  Transforms the expression expr by giving each summation and product a unique index. This gives `changevar` greater precision when it is working with summations or products. The form of the unique index is `jnumber`. The quantity number is determined by referring to `gensumnum`, which can be changed by the user. For example, `gensumnum:0$` resets it.

  Categories:Sums and products ·

Function: lsum (expr, x, L)

  Represents the sum of expr for each element x in L. A noun form `'lsum` is returned if the argument L does not evaluate to a list.

  Examples:

      (%i1) lsum (x^i, i, [1, 2, 7]);
                                  7    2
      (%o1)                      x  + x  + x
      (%i2) lsum (i^2, i, rootsof (x^3 - 1, x));

                           ====
                           \      2
      (%o2)                 >    i
                           /
                           ====
                                         3
                           i in rootsof(x  - 1, x)

  Categories:Sums and products ·

Function: intosum (expr)

  Moves multiplicative factors outside a summation to inside. If the index is used in the outside expression, then the function tries to find a reasonable index, the same as it does for `sumcontract`. This is essentially the reverse idea of the `outative` property of summations, but note that it does not remove this property, it only bypasses it.

  In some cases, a `scanmap (multthru, expr)` may be necessary before the `intosum`.

  Categories:Expressions ·

Option variable: simpproduct

  Default value: `false`

  When `simpproduct` is `true`, the result of a `product` is simplified. This simplification may sometimes be able to produce a closed form. If `simpproduct` is `false` or if the quoted form `'product` is used, the value is a product noun form which is a representation of the pi notation used in mathematics.

  Categories:Sums and products ·Simplification flags and variables ·

Function: product (expr, i, i_0, i_1)

  Represents a product of the values of expr as the index i varies from i_0 to i_1. The noun form `'product` is displayed as an uppercase letter pi.

  `product` evaluates expr and lower and upper limits i_0 and i_1, `product` quotes (does not evaluate) the index i.

  If the upper and lower limits differ by an integer, expr is evaluated for each value of the index i, and the result is an explicit product.

  Otherwise, the range of the index is indefinite. Some rules are applied to simplify the product. When the global variable `simpproduct` is `true`, additional rules are applied. In some cases, simplification yields a result which is not a product; otherwise, the result is a noun form `'product`.

  See also `nouns` and `evflag`.

  Examples:

      (%i1) product (x + i*(i+1)/2, i, 1, 4);
      (%o1)           (x + 1) (x + 3) (x + 6) (x + 10)
      (%i2) product (i^2, i, 1, 7);
      (%o2)                       25401600
      (%i3) product (a[i], i, 1, 7);
      (%o3)                 a  a  a  a  a  a  a
                             1  2  3  4  5  6  7
      (%i4) product (a(i), i, 1, 7);
      (%o4)          a(1) a(2) a(3) a(4) a(5) a(6) a(7)
      (%i5) product (a(i), i, 1, n);
                                   n
                                 /===\
                                  ! !
      (%o5)                       ! !  a(i)
                                  ! !
                                 i = 1
      (%i6) product (k, k, 1, n);
                                     n
                                   /===\
                                    ! !
      (%o6)                         ! !  k
                                    ! !
                                   k = 1
      (%i7) product (k, k, 1, n), simpproduct;
      (%o7)                          n!
      (%i8) product (integrate (x^k, x, 0, 1), k, 1, n);
                                   n
                                 /===\
                                  ! !    1
      (%o8)                       ! !  -----
                                  ! !  k + 1
                                 k = 1
      (%i9) product (if k <= 5 then a^k else b^k, k, 1, 10);
                                    15  40
      (%o9)                        a   b

  Categories:Sums and products ·

Option variable: simpsum

  Default value: `false`

  When `simpsum` is `true`, the result of a `sum` is simplified. This simplification may sometimes be able to produce a closed form. If `simpsum` is `false` or if the quoted form `'sum` is used, the value is a sum noun form which is a representation of the sigma notation used in mathematics.

  Categories:Sums and products ·Simplification flags and variables ·

Function: sum (expr, i, i_0, i_1)

  Represents a summation of the values of expr as the index i varies from i_0 to i_1. The noun form `'sum` is displayed as an uppercase letter sigma.

  `sum` evaluates its summand expr and lower and upper limits i_0 and i_1, `sum` quotes (does not evaluate) the index i.

  If the upper and lower limits differ by an integer, the summand expr is evaluated for each value of the summation index i, and the result is an explicit sum.

  Otherwise, the range of the index is indefinite. Some rules are applied to simplify the summation. When the global variable `simpsum` is `true`, additional rules are applied. In some cases, simplification yields a result which is not a summation; otherwise, the result is a noun form `'sum`.

  When the `evflag` (evaluation flag) `cauchysum` is `true`, a product of summations is expressed as a Cauchy product, in which the index of the inner summation is a function of the index of the outer one, rather than varying independently.

  The global variable `genindex` is the alphabetic prefix used to generate the next index of summation, when an automatically generated index is needed.

  `gensumnum` is the numeric suffix used to generate the next index of summation, when an automatically generated index is needed. When `gensumnum` is `false`, an automatically-generated index is only `genindex` with no numeric suffix.

  See also `lsum`, `sumcontract`, `intosum`, `bashindices`, `niceindices`, `nouns`, `evflag`, and zeilberger-pkg

  Examples:

      (%i1) sum (i^2, i, 1, 7);
      (%o1)                          140
      (%i2) sum (a[i], i, 1, 7);
      (%o2)           a  + a  + a  + a  + a  + a  + a
                       7    6    5    4    3    2    1
      (%i3) sum (a(i), i, 1, 7);
      (%o3)    a(7) + a(6) + a(5) + a(4) + a(3) + a(2) + a(1)
      (%i4) sum (a(i), i, 1, n);
                                  n
                                 ====
                                 \
      (%o4)                       >    a(i)
                                 /
                                 ====
                                 i = 1
      (%i5) sum (2^i + i^2, i, 0, n);
                                n
                               ====
                               \       i    2
      (%o5)                     >    (2  + i )
                               /
                               ====
                               i = 0
      (%i6) sum (2^i + i^2, i, 0, n), simpsum;
                                    3      2
                         n + 1   2 n  + 3 n  + n
      (%o6)             2      + --------------- - 1
                                        6
      (%i7) sum (1/3^i, i, 1, inf);
                                  inf
                                  ====
                                  \     1
      (%o7)                        >    --
                                  /      i
                                  ====  3
                                  i = 1
      (%i8) sum (1/3^i, i, 1, inf), simpsum;
                                      1
      (%o8)                           -
                                      2
      (%i9) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf);
                                    inf
                                    ====
                                    \     1
      (%o9)                      30  >    --
                                    /      2
                                    ====  i
                                    i = 1
      (%i10) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf), simpsum;
                                        2
      (%o10)                       5 %pi
      (%i11) sum (integrate (x^k, x, 0, 1), k, 1, n);
                                  n
                                 ====
                                 \       1
      (%o11)                      >    -----
                                 /     k + 1
                                 ====
                                 k = 1
      (%i12) sum (if k <= 5 then a^k else b^k, k, 1, 10);
                10    9    8    7    6    5    4    3    2
      (%o12)   b   + b  + b  + b  + b  + a  + a  + a  + a  + a

  Categories:Sums and products ·

Function: sumcontract (expr)

  Combines all sums of an addition that have upper and lower bounds that differ by constants. The result is an expression containing one summation for each set of such summations added to all appropriate extra terms that had to be extracted to form this sum. `sumcontract` combines all compatible sums and uses one of the indices from one of the sums if it can, and then try to form a reasonable index if it cannot use any supplied.

  It may be necessary to do an `intosum (expr)` before the `sumcontract`.

  Categories:Sums and products ·

Option variable: sumexpand

  Default value: `false`

  When `sumexpand` is `true`, products of sums and exponentiated sums simplify to nested sums.

  See also `cauchysum`.

  Examples:

      (%i1) sumexpand: true$
      (%i2) sum (f (i), i, 0, m) * sum (g (j), j, 0, n);

                           m      n
                          ====   ====
                          \      \
      (%o2)                >      >     f(i1) g(i2)
                          /      /
                          ====   ====
                          i1 = 0 i2 = 0

      (%i3) sum (f (i), i, 0, m)^2;
                           m      m
                          ====   ====
                          \      \
      (%o3)                >      >     f(i3) f(i4)
                          /      /
                          ====   ====
                          i3 = 0 i4 = 0

  Categories:Sums and products ·Simplification flags and variables ·